That's probably more of an issue for the Whites than the Reds: If confronted with groups of semi-anarchist Mountain Man types who just want to be left to their own devices, the latter are likely to just shrug and say, "Fine by us, don't make trouble and we'll stay out of your hair."
In an unposted speculation on the dynamics in West Virginia I hit on that guess myself; the Reds have little to lose and much to gain by permitting de facto autonomy in peripheral regions provided they won't impede the extraction of vital resources--if push comes to shove they can regard populations that are reluctant to permit even that as sitting on strategic reserves and turning to other sources, pushing the matter if and only if the cost of forgoing exploiting the blocked reserves is too high.
There never was nationwide Prohibition in this TL either, so the conflicts over booze have been restricted to local venues, state by state, and the states that have gone as far as to ban it (generally on a county by county basis at that) are those where there is widespread popular support for the idea of prohibition. This is probably very hypocritical, analogous to puritanical notions in general--people want to be on public record as supporting the "right thing to do" while in practice many of them cheat privately one way or another. But Appalachia has not been treated to the OTL phenomenon of moonshiners versus G-Men; illegal moonshiners there may be, since these states are just those which would be most in the grip of teetotal moralizing--indeed if there are several breakaway Autonomous Zones in West Virginia, other Appalachian regions and the Ozarks, I bet on paper their recognized governments are quite Bible Belt in their formal rules, and rife with individual "sinners" who voted for these rules even as they violate them. With these zones abstracted from the default regional governments (I forget whether the default carry over from US states are called "states" or what) the latter are likely to shift toward the national norm considerably.
The Reds in hillbilly country care mainly that their resources, including as transport routes, be denied to the Whites and that they be made available to the Red cause; with a democratic majority well in hand nationally their concern for the local politics of these small population regions will mainly be concerned just to make sure they are not harboring and cultivating violent reactionaries.
Mind the Reds will include some people who by background or even by current residence and occupation are native to these regions! That's a double edged sword of sorts--on one hand, it will ease negotiations and communications between the national Red movement and the dissident peoples of these regions. On the other, these individual Reds will have scores to settle locally that might distort national policy both as they skew intelligence from the regions and as they try to carry out personal vendettas. It is a question at that point of party discipline and self-discipline whether the Party can discern such hotheaded individuals, make allowances for them or just reroute them elsewhere, and whether such persons can subordinate their personal agenda to the larger picture.
Secondarily the do-gooder aspect of radicalism will be concerned to make sure no one is being unjustly exploited--if a particular autonomous zone is liable to oppress African American settlements, for instance, either the latter get their own AZ with a right to defend themselves or are simply included in the larger bailiwick of the legacy "state" or whatever it is called.
There are enough women and is enough feminist consciousness even in this first Red generation that concerns about the status of women in these zones will result in do-gooders poking their noses in and making sure the women and girls there are not being treated too badly, and can leave if they feel terrorized, and I suppose in a few cases, regional zones that seek to sit out the massive Red transformations will be found to be too reactionary and be broken up.
In this context note that in the longer run, after the civil war, the UASR is rich enough to have a strategy of recruitment of these regional peoples by generous social aid without demanding equal compensation, "from each according to their abilities to each according to their needs!" De facto the Party might wind up subsidizing backward bourgeois norms in their populist Jeffersonian form via welfare aid relieving a fundamentally capitalist regional economy of its burdens, and via social services such as support for education--which is also, handled diplomatically, a pretext to inject Red ideology in the form of schoolteachers, both from outside the region and by training up more generations of teachers in UASR level supported local teaching colleges.
I imagine factions of the national party will point out the contradictions of not confronting local society with the hard choice between Red evolution versus suffering the full costs of ongoing capitalism, and denounce such compromises. But pragmatically it is a question of scale in terms of population whether it is costly to the proletarian movement to baby these backwaters in this way, and I think pragmatically it won't cost much and the benefits, both in terms of immediate peace and reconciliation of otherwise thorny regions and the gradual recruitment of these regions to a Redder point of view will start manifesting pretty plainly. From the beginning, even regions demographically far from the norms that give rise to normal levels of Worker Party support will still have some Reds native to them as well as imported Reds working their way into the fabric of local society after all. These people can form the nuclei of a regional party cadre that will steadily rise in proportion and in absolute numbers.
But during the Civil War the Reds can afford to kick the can down the road if such a region will agree to forego aiding the Whites and make agreements with people they might regret having made later. Any subsequent revisiting of these informal treaties later formalized under UASR reorganization will happen in the context of due process of law in the context of course of strong Red political domination. To an extent, reaction that Reds dislike will be covered by the more conservative minority parties...but the latter are walking a tightrope between maintaining conservative dissents from the Red enthusiasm, and being seen as aiding downright reaction. The distinction is pragmatic, though once the UASR period proper begins formal revised common law will seek to draw some rational lines of principle between culpable reaction and tolerable conservatism. So, zones that are full of people who have misgivings about the Red program across the board will need to seek the umbrella protection of the tolerated conservative parties--I am not forgetting that the terminology of the early UASR rejects the term "conservative" as acceptable, but translating into terms more characteristic of OTL here.
In terms of their subjective consciousness of self interest, indeed we can envision large regions that one might assume would dissent from the sweeping Red program. But even these have large majorities who had OTL and here also specific grievances against the capitalist order. OTL the power of capital was too strong to simply oppose directly and check; people had to learn to make their peace with a basic order with aspects quite oppressive to them and rationalize it as they could.
I have in mind the OTL book
White Collar by the sociologist C. Wright Mills, particularly the analysis of the transformation of the USA from a rural centered agrarian capitalism which the customary OTL ideology of the virtues of the free market society and the "natural aristocracy" of business successes evolved to suit. The habit we have in these late generations far removed now from the great transition from an overwhelmingly rural nation to urban-industrial of thinking of country people as inherently reactionary relates in part to the bitter discontent the industrial-capitalist liquidation of this former Jeffersonian Arcadia left in its wake. Prior to being broken and subordinated by centralized capital, which is the everyday reality of the modern USA and to which the powerless people of the peripheral zones respond to with all sorts of ideological evasions and projections, straightforward organized resistance was part of the very stuff of such movements as the People's Party. The Klan and other reactionary activists had a complicated relationship with frustrated populism, being from some analytic points of view the mere tools of the very centralizing order that so alienated them but from others being an expression of the anger and discontent of a portion of the alienated sector; I think this describes right wing faux-populism in general.
But in this TL, the mighty struggles of the 1880s-1910s are still in living memory, and the Debs-DeLeonists and other fellow traveling left radicals are widely seen as heirs to the legacies of many of these groups. People whose narrow sectional interest does not overlap the full Worker's Party agenda still share some sense of partial relationship and so it is much more thinkable that a narrower isolationist populism would see alliance with, or anyway friendly neutrality with respect to, the Reds as natural and the Reds as reasonable negotiators in good faith. Of course reactionary interests will have attempted to ideologically drive wedges between churchgoing hillbillies or prairie farmers or western mountain-Great Basin populations, but the Red base is broad enough that people who were easily convinced of the devilish nature of the Reds OTL will at least reserve judgement and decide based on specific negotiations. In turn the Reds will have a broad enough base that deeply committed Debs-DeLeonists will include people from these backgrounds in enough numbers that they always have someone handy fairly high up in Party circles who speaks their language and will talk turkey with them.
The deal the UASR can initially offer the people whose very foundations of life as they understood it were being drained away by global capitalist mechanisms, many of them anyway, is a truce in which they are encouraged to be autonomous and can maintain their older ways for a time on tolerance, while the best resources for rationally and systematically feeding the industrialized Red masses and supplying intensive industry with raw materials are found on lands that have long ago been alienated to highly centralized ownership. So if some efficiency relative to an imaginary case where 100 percent of the working masses are fully on board is lost, so be it; the rich UASR can afford the sacrifice if it buys victory in the civil war and peace in the countryside.
Gradually the peoples initially reluctant to get behind the Red bandwagon will start slipping into that parade, one by one, voting with their feet and then, as people with stronger ties and opportunities holding them to the land in turn come to be attracted to Redder ways of doing things, the remainder will start voting formally in place more and more Redly. Individual enterprises and expedient policies will gradually weave even the most autonomous zones into the national and indeed Comintern wide global fabric, and any discontents and conflicts this process brings will get a fair public hearing and explicit compromises will be brokered. But gradually they too will shift Redward, perhaps retaining some peculiar regional distinctions...for instance perhaps remaining bastions of Christian religious faith. But that would be far less a marker of a general reactionary agenda than it is OTL.