Speaking of tech, this is about to be adopted. Maybe with earlier war experience it can get up to Kar98k standard earlier, or even get a 10 round magazine Lee-Enfield style.
Edit: Or if you want a cooler action, the Schmidt-Rubin straight pull from Switzerland. Maybe a few for the snipers to supplement the theoretical super Mauser?
I suspect things will probably be delayed on our end; if the Gewehr 88 works, I don't think we'd want to interrupt production for a new rifle (likely only making superficial modifications such as, say, simplifying production or reducing overall length). I'd imagine it would make most sense for us to take stock of things once the war's over and try and develop a wholly new rifle based on combat experience.
On the topic of the Schmidt-Rubin, we'll presumably be capturing large numbers of Mannlicher M1888 rifles which are also a straight-pull action. I suspect we'd have a significant amount of experience fighting against them, and knowing how things went IRL, German soldiers will use captured guns as well if they like them.