So, there are a number of things I think are important to go over that Thompson would know, understand, and seek to implement with the time and influence available to him.
1) As has already been discussed, the Thach Weave. What's not nearly as well known is that the Weave was actually an excellent tactic for escorting squadrons of bombers, too--indeed, this was successfully used at Midway, where a pair of Wildcats performed the Weave right over a squadron of torpedo bombers, managing to give them astonishingly good protection against a handful of Zeros, until yet more Zeros arrived and overwhelmed the formation (forcing it to split up, somewhat). Thach himself advocated for this tactic, and though I'm uncertain if it was effectively implemented later, it was nonetheless sound in theory. One of the problems 1942 USN carrier air groups struggled with was providing effective cover for the bombers, particularly when the torpedo bombers were so slow. The Thach Weave prevented formations from splitting up or not even realizing one another were under attack (as happened at Midway), since the fighters were right there with the bombers. It also forced enemy fighters to deal with the fighters first, because the Wildcats would be right there on top of the bombers, preventing effective diving attacks at the bombers. Notably, Thompson could have shown various Army squadrons stationed at Pearl during peacetime the Weave, teaching them it, giving it popularity in the Army Air Corps and having ample time to spread.
Additionally, one of the main benefits of the Weave was not merely the ability to brush enemy fighters off of a fellow fighter's tail, but to be able to watch each other's backs even when not engaged, and to be able to communicate an incoming fighter attack immediately by turning towards your counterpart across from you. He would see the other fighter turning towards him, and know to immediately mirror the maneuver, even if he had no idea there were enemy fighters or where the enemy was coming from. By positioning two fighters--a leader and a wingman--directly across from each other, they could watch each other's backs with ease. Thach also advocated for the four-fighter element, subdivided into two pairs. While this was already in practice, some were of a mind that it should be six-fighter elements, subdivided into three pairs, but this proved to be unwieldy.
2) CAG composition. It took time and hard-earned experience for the USN to realize that their carrier air group compositions were flawed--they had too many torpedo bombers and not enough fighters. While Thompson might not have the clout to set the doctrine for the other carriers (though maybe he managed to convince them with that exercise/wargame?), he could adjust his own air group. Even if he couldn't get the fighters he wanted, he could almost certainly swap out some of his torpedo bombers for dive bombers, and just have said dive bombers not be loaded down with bombs, and keep them for defense of the carrier. This is for several reasons:
A) Until the Avenger came along, American torpedo bombers were horrendously slow. So much so, in fact, that making effective torpedo attacks against Japanese carriers (to say nothing of cruisers) was absurdly difficult. Unlike in World of Warships, torpedoes of the day were--at best--barely faster than their targets, meaning that even a carrier could just turn away from torpedoes and just outrun them until they ran out of fuel. Thus, you had to cross-drop them from both sides simultaneously, which means overtaking the enemy carrier even as it turns constantly. Needless to say, speed is paramount for a torpedo bomber to create a proper setup for such an attack. Until Avengers come along, it made more sense to use torpedo bombers as a way to finish off ships crippled by dive bombers.
B) One thing the USN realized after Coral Sea--and especially after Midway--was that fighter cover was the only reliable means by which a carrier could survive enemy air attack. And fighter cover was essential for conducting successful attack against carriers as well.
3) Thompson would have had enough time to pressure Grumman (or get his superiors to pressure Grumman) to keep the Wildcats at a four-gun configuration instead of a six-gun config, particularly since operational range is paramount for carrier fighters, and the Wildcat was already overweight--adding two more guns weighed down the plane even more.
4) He could also have told Grumman of the pressing need for effective drop tanks for the Wildcat, to extend its range. While these were sort of available prior to Midway, they were more jury-rigged than anything, and prone to leaks and various other problems.
5) Thompson did prove that carrier warfare was heavily influenced by a numbers advantage--having more carriers than your enemy changed the game radically. However, one thing he would know is that having multiple carriers operating nearby to each other meant that they could effectively share their CAP, coordinate with each other, and increase their radar coverage (less chance of radar missing something if there are two different sets of radar slightly apart searching the same area).
6) Drop tanks for the dive bombers. Since the Dauntless was the scouting plane of choice for American carriers, getting the most range and loitering time out of them as possible is paramount. He could have called for large drop tanks to be made for the Dauntless for this purpose.
7) Using USN submarines not to form a cordon, but to be given sectors to search for the enemy and report their positions before attacking. This was a major mistake at Midway--all but two of the USN's submarine were stationed to form a cordon where the Japanese were unlikely to be anyway, and where allied scouting planes could more effectively cover that area regardless.
8) Impressing the importance of dive bombing over glide bombing to the Army Air Corps and Marines. Namely, that glide bombing is extremely ineffective and outright dangerous to the air crews that try it. Insisting that any AAC air crews that fly dive bombers learn how to dive bomb and always attempt to dive bomb over glide bombing.
9) Suggest that B-17s that are assigned to attack enemy carriers try to draw out their attacks as long as possible, rather than expend all of their bombs as quickly as possible. This is because high-altitude bombing is very unlikely to hit, but it forces the enemy carriers to dodge, which prevents them from landing, launching, or spotting aircraft for the duration of the attack. Thus, dropping bombs sparingly buys the allied carrier force lots of time and initiative. This was taken advantage of at Midway, but could have been even more effective had it been doctrine to draw out such attacks as long as possible.
10) The importance of sailing relatively erratically when the weather is too poor to launch inner air patrols (to find submarines close by). This would obviously be very important to Thompson, stationed aboard Saratoga, the ship that repeatedly got torpedoed at very inconvenient times.
11) The Japanese had a thoroughly ingrained tendency to make absurdly complicated and intricate operational plans. While this worked out well for them when they had tons of time to prepare, discover issues and come up with solutions, and had the full initiative, it backfired horrendously almost all other times, such as at Coral Sea, Midway, Guadalcanal, Leyte Gulf, the Philippine Sea, etc. This was a trademark of Yamamoto, certainly, but it was a tendency shared by all Japanese commanders, particularly in the IJN. They also suffered heavily from "plan intertia", which meant that officers and commanders tended to go along with a plan even when they knew it was impossible to carry out (or that circumstances had rendered the premise of the plan moot).
12) USN doctrine for carriers facing enemy air attack was to save maneuvering until as late as possible, and only when necessary, so as to avoid spoiling your own AA's firing solutions or scattering your defensive formation. While this made sense in 1944, in 1942, AA firepower and tech is not nearly good enough to make this reliably effective, especially against dive bomber attack. Thompson might communicate this to his fellow carrier captains/admirals.
13) In 1942, the Japanese had a tendency to split up their naval forces to accomplish many objectives at once (or within a short time span), which led to disastrous results at Coral Sea and Midway. This, combined with the Japanese doctrine of "distant cover", wherein ships would be very far apart, often too far apart to help each other when they needed it, could be exploited, particularly by ABDACOM.
14) Japan had a really hard time replacing its carrier pilot losses with pilots of comparable quality. After the major losses they took at Pearl Harbor, their performance would be significantly poorer, going forward.
15) Japan's biggest problem with its conquest of the Southwest Pacific was logistics. Even when they conquered Singapore, they were on their last legs, logistically, and were barely in any better shape than the force they defeated there. With working, reliable (or at least substantially more reliable) torpedoes, American submarines might be able to make a significant impact on Japanese logistics, in turn hampering Japanese advances in the region.
16) Japanese AA was notoriously bad at shooting down American dive bombers. During the entirety of Midway, Japanese AA shot down a grand total of one dive bomber. By contrast, American AA shot down at least two (including one before it even dropped its bomb, while still in its dive), despite being attacked by dive bombers only once, and with far fewer than the Americans attacked even Kaga alone with. Japanese AA guns on carriers tended to have poor firing arcs (due to poor placement, and thus lots of obstruction) and visibility. Also, Japanese carrier formations were very spread out, which meant that not only were the carriers largely on their own for AA defense, but everything short of heavy cruisers (in the IJN) tended to have very little in the way of AA firepower in 1942.
17) It wasn't until after the lessons learned at Midway that the IJN took conducting good searches with its scout planes seriously. Indeed, launching any of a carrier's own planes for scouting was against doctrine (with the sole exception of a prototype D4Y meant for scouting, at Midway).
18) Having proven that numerical superiority is extremely important in carrier warfare, it's possible--especially with him having the president's ear for so long--that Thompson convinced Roosevelt to reassign all but Ranger to the Pacific. Given that none of the pre-North Carolina battleships had the speed to keep up with carriers, he could propose transferring all American battleships (including the ones that only need minor repairs, like Pennsylvania) to the Atlantic and/or Mediterranean, freeing up British carriers to operate in the Mediterranean or elsewhere.
19) Thompson would know that assigning long-range bombers/patrol craft to cover the Atlantic Gap ASAP would save millions of tons of merchant shipping, and that the British were astoundingly slow to realize this and do something about it.
20) Thompson would know that it took the US far too long to realize that implementing a convoy system for its merchant shipping (and having destroyers directly escort convoys instead of patrolling designated areas by themselves) would prevent horrendous losses in merchant shipping. His time with Roosevelt could have allowed him to impress upon him the importance of implementing these measures in case the US ever found itself at war with Germany--and faced the threats of U-boats. He could point to World War 1 as the definitive evidence of this--once the British implemented a convoy system, shipping losses dropped dramatically.
21) I know I've brought this up before, but armor-piercing and semi-armor-piercing bombs for dive bombers. The Japanese used these to great effect, and they're kind of a common-sense weapon. Basically, you take a shell of an appropriate caliber (cruiser- to battleship-sized), adapt it for mounting onto a dive-bomber, and viola! You have a dive bomber capable of dealing a perfect, crippling blow to a ship's machine spaces/engineering/boilers, potentially even doing damage to the hull below the waterline (depending on the target and caliber of shell used, plus luck). You can bet your ass that, when Thompson proposes the idea to other carrier admirals/captains, they'd jump all over that--a 1000-pound HE bomb isn't going to do crippling damage to a battleship, for one, and an HE bomb isn't going to do fatal damage to a carrier unless it happens to start unending fires. But an AP bomb could easily knock out a ship's boilers, or its engines, or even its magazines, making it easy prey to follow-up attacks, especially with torpedo bombers.