Hm... what if the Republicans give in to their progressive wing in the teens and twenties?
Have Roosevelt win the 1912 nomination and win the Presidency, and push hard for the GOP to become progressive. Have LaFollette continue doing this throughout the 20s, so that the GOP becomes the more liberal of the two parties. Then, have everything proceed at OTL until the 60s. Once the Democrats swing in favor of civil rights, along with the progressive GOP (unlikely, admittedly, in this scenario), have the Dixiecrats stay around, and gradually "moderate" (drop segregation, basically), and then come roaring back in the 80s. The GOP becomes a New England regionalist party, occasionally winning plains states if they get lucky. They generally caucus with the Democrats in the House and Senate. The strong support in the north keeps them from being completely wiped out as a party.
Admittedly, this scenario is probably a stretch, but that's how I think it could happen.