Some ideas:
Have union power remain strong in America.
Curtail the rise of the New Left to some degree. Or perhaps have a less of a reliance on college students and/or have it develop in a slightly different direction.
Keep the New Deal Collation together and have some of the social issues seep into the rest of the Collation over time.
Lessen (or altogether stop) the US's involvement in the Vietnam War.
Bobby Kennedy doesn't get shot in California and defeats Richard Nixon in '68 (Kennedy might not win the Appalachian region, but this could stop the Southern Strategy and/or Silent Majority philosophy dead in its tracks).
The main one though might be lessening the rural/urban divide that has risen in America. At the very least, stop the resentment (justified or not) that is felt in Rural America toward others. Not sure
how to do it, mind. But getting rid of that resentment is a good way of stopping identity politics and wedge issues from developing.
Since the PoD can be as far back as the Civil War, the best bet might be to play with Reconstruction Politics. But that's another section of the board.

Failing that, I think butterflying away the Vietnam War is the next best bet. This might keep the New Deal Collation together longer, as well as not providing much of the fuel that lead to the New Left.
You'd still have the Sexual Revolution to contend with, however. So that might throw a spanner in the works.
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Going a bit further afield, given that much of the Appalachians are centered around mining, you might have some sort of responsible use environmental movement arise from
within the Appalachian community. If that gets nursed the right way, they could make common cause with people concerned with worker safety, especially if unions remain potent. This strengthens the bond with other left wing concerns.