the right-wing part of the political spectrum, in general those who were members of the Republican Party.

Just so you know, this wasn't true back then. From the 1830s to the at least FDR the parties were the other way around. Democrats were the party of conservatives and the South, Republicans were the progressive northern party. It was the party of Lincoln after all. Republicans used to be pro-union, pro-workers rights, pro-women's suffrage, while Democrats were the party of the Klan and Jim Crow.

The shift started under FDR when the progressive wing of the Democratic Party began to grow to match the progressiveness of the Republican party. FRD kept the South because he was a Democrat, but expanded to northern cities by being liberal which the Democrats normally were not. Then Truman went and desegregated the military. The south ignored it when FRD did things that helped northern blacks, but considered a racially integrated military a bridge too far. After trying to snatch the nomination from Truman, they simply broke away and started their own Democratic party the Dixiecrats, one that returned to the traditional conservative roll the party had.

FRD had successfully moved the conservative Democratic party to be both conservative and progressive, but all Democrats after him found it impossible to win both and each sided with the liberal north over the conservative South. Eventually, that power vacuum in the south lead to the "Southern Strategy" of Nixon who brought the once Democratic conservatives into the Republican party making the Republican party the new home of conservatives.

But back in the 1920s, it was the Democrats for the most part who were the conservatives.