The 1920s were, overall, some of the most contested primaries in a fair long time. The Democrats resorted to dark horse candidates (James Cox in 1920 and John W. Davis in 1924) to break progressive/conservative feuds. In 1920, so did the Republicans; only Harding's personal popularity and his endorsed successor kept the Republican Party in its straight line. Coolidge ran almost unopposed in 1924, receiving a direct majority in a three-way race, but let's remember: he was challenged for that nomination to begin with.
So let's pretend Coolidge repeats his 1928 stunt and decides to retire back to New England. Who runs, on which side? Hiram Johnson? Frank Lowden? Does Robert La Follette make a good shot at a third party?
I've been interested in 1920s situations before (Wood) but this seems ripe for potential.