It started flipping in the 1960s, but Southern Democrats hung on to most southern state governments, and many House & Senate seats, until the 1990s. Richard Shelby (a conservative Alabama Senator) only switched parties in 1994, for example.
In 1952, Robert Taft wanted to run for President using the "southern strategy" to woo disaffected Democratic voters, but Eisenhower decided to run, and that put an end to that strategy until Nixon was more successful in 1968.
Basically, the south became squishy on Democratic Party once northern Democratic Mayors (and FDR) realized that African-Americans were a strong labor voice and a powerful voting block. "The Party of Lincoln" (GOP) had been ignoring African-Americans since TR was in the White House.