Well first and foremost you need to destroy the Democratic Party. They've been the bulwark of Jim Crow on a state level, and breaking them, along with preventing the States Rights Act getting passed (not impossible, LBJ had to use every dirty trick in the book to get it passed in the first place), is absolutely required to allow desegregation to go through. IOTL even during Republican administrations desegregation was only limited to the armed forces and Federal employees, with the GOP unable, and let's be honest often unwilling, to challenge the State powers that upheld Jim Crow, even under outspokenly pro-civil rights Presidents like McCarthy or Goldwater.
Whilst it is true that there were anti-segregation elements in the Democratic Party, and some have suggested having them become dominant in the party, I think that's unlikely. A more realistic scenario would be to have Henry Wallace's Social Democratic Party somehow supplant the Democrats, but even then that's a long shot that would require preventing both the Second and Third Red Scares, WW3, and the post-war anti-leftist reaction in the wake of the fall of the Soviet Union. The destruction of the Social Democrats effectively destroyed the anti-segregation wing of the Democrats along with the more actively pro-civil rights Republicans, and is part of the reason why it wasn't until the mid-80s that desegregation was back on the agenda for American progressives (and there are still plenty of people trying to make the progressive case for "separate but equal").
To be perfectly blunt, short of a revolution I don't see it happening. White supremacy is just too heavily entrenched in the US political system to allow it to reform away.