Maybe in Plessy v. Ferguson (1896), the Supreme Court decides on more limited grounds that the railroad in question was technically private.
The direct wrongfulness of the decision combined with, wow, railroads receive a lot of direct and indirect government benefits to be called fully private, gets the ball rolling. In the '00s and '10s, there's a variety of state laws that common carriers will operate in a nondiscriminatory manner. In the '20s, state laws that employment will be nondiscriminatory. In the '30s, New Deal programs.
Combined in the '30s with laws and actual implementation that enlistment and promotion in the U.S. military will be nondiscriminatory. And during WWII, the vast majority of units are nondiscriminatory. The occasion segregated unit is viewed as an anachronistic throwback, almost as quaint, and as an inefficient use of personnel.
And that's the linchpin which changes everything. Once African-American citizens have served in significant numbers, there's not much more case left for anything but full equality.
PS All the above time sequence is one way I'd hope it would work out. But a lot of state law became ever more discriminatory as it's a way for crappy stare legislators to appeal to the masses. So, maybe the first thing to do is a Reconstruction which is serious about the voting rights of persons newly freed from slavery.