1920: Harding wins Missouri, West Virginia, Oklahoma, and Tennessee
1924: Coolidge wins Missouri, West Virginia, Oklahoma, and Kentucky
1928: Hoover wins Missouri, West Virginia, Kentucky, Tennessee, North Carolina, Arkansas, Florida, Oklahoma and Texas.
He also is within half a point of winning Alabama.
1952: Eisenhower wins Missouri, Virginia, Tennessee, Florida, Texas, and Oklahoma.
1956: Eisenhower wins Kentucky, Tennessee, Louisiana, Florida, Texas, Oklahoma, Virginia, and West Virginia.
The GOP had already proven to be competitive in the South before the passage of the Civil Rights Act. Odds are the GOP will continue to make some gains down south as urbanization-suburbanization and sunbelt migration come into play. The Southern GOP would be an amalgamation of suburban interests, black interests, and business interests.
Goldwater might end up nominee in 1968, but any Civil Rights Act Nixon puts through would probably be a bit more moderate, so I imagine Goldwater would have voted for the CRA TTL. But insofar as just how moderate the 1964 law would be, Goldwater was on board with most of it but disagreed with Title II (desegregation in places of public accommodation) on freedom of association grounds - the impression I get is that the public was ready for a significant Civil Rights Bill by the mid-to-late 1960s. A more moderate law could perhaps mean less backlash.