Perhaps an early death of FDR, or a strong influence of the conservative wing of the Democratic Party, weakening FDR and the Progressive wing of the Democratic Party, disappointing a certain number of progressive Democratic voters and convincing them to (re-)back the Progressive wing of the Republican Party.
A strong support of local African-American candidates by the Republican Party, to emphasize its heritage of Abolitionism, and a greater opening to the immigrant communities, all laboriously done for a decade, might help.