The depression hit the South hard and public works kept a lot of folks there from starving. By the end of his second term though, FDR's progressive agenda was clearly backfiring but Southern voters kept on supporting it.
Interestingly, Long was said to have been considering a run for President himself in 1936, before getting shot by the son of a rival.
Yes, FDR wasn’t liberal/socialist enough for some Southern Democrats - there was a lot of support for Huey Long to run for President. “Share Our Wealth” was their motto.