> So they started voting for the people responsible for the KKK and Jim Crow laws
No, the people who still supported the KKK and the ideology behind Jim Crow laws switched major parties, if they were associated with a major party, from the Democrats to the Republicans between the late mid-1960s and the mid-1990s, because of disaffection with the Democrats (initially particularly President Johnson) backing civil rights, but also because the Republicans started actively campaigning to their racist interests to leverage their disaffection with the Democrats.
In some cases, the people switching parties were literally the same people who had bolted temporarily from the Democratic Party in the 1940s over Truman's integration policies, but who had come back because they had no other major party to go to, and their own separate party had failed. (Strom Thurmond, the Dixiecrat Presidential candidate in 1948, is a notable, highly-visible example.)
And as this process continued it was self-reinforcing, because the more White supremacists left the Democratic Party for the Republicans, the stronger the Democratic support for the interests White supremacists opposed could be.