In my opinion, things here in the TL are at a critical juncture. True, the late 19th Century saw the solidification of Social Darwinism, a worldview that saw nothing but conflict, either at pause or ongoing, between different groups of people, but that was encouraged in order to give a coat of seemingly scientific legitimacy to practices that had already consolidated (segregation and race-based colonialism, and the book that "scientific racists" would use to push Social Darwinism, The Origin of Species, is still only a few years old, a just recent arrival on the shores of America. Herbert Spencer first uses the phrase "survival of the fittest" (more charged and a bit more misleading than Darwin's preferred "natural selection") in 1864. A North that has much more experience with whites and blacks working together, is less likely to reconcile with intellectuals from the South, has four more years of Lincoln, and has the prewar Southern power structure more thoroughly shattered and discredited, could possibly roll down the other side of the hill, formulating notions about the capacity for populations to learn to tolerate each other that IOTL had to wait until the 30s or 40s, long after when segregation had embedded itself deep back into the South. What is important, in my view, is the measures adopted don't have to be particularly radical to create a snowball effect; it could be efforts so moderate (just intolerant of actively reimplementing subjugation of blacks under whites) that the effect of breaking down racism could be done unintentionally by people in power who don't particularly find the idea of racism to be a problem. Just putting people of different races into the same spaces, and making sure it remains reasonably civil, will be enough to start causing a change.
Schools have a lot of potential, but I think they'd need to be boarding schools to get full results if efforts aren't made to deradicalise adults as well. Otherwise, children with racist parents would still be able to go home at the end of every day to receive a "topping up" of the indoctrination they were given before they started school. Also, teenagers/young adults at the best of times still aren't the best at emotional maturity, and the ones in these schools would be dealing with either racist indoctrination or the trauma of having been human property from birth and until recently, so teachers would likely have to be mindful and interventionist to deescalate conflicts and stop the forming of gangs.
Another interesting issue is one of parasocial relationships. Contact between whites and blacks doesn't even have to be universally physical, it can also be by the connections a person forms to others without it being the other way around. Athletes, entertainers, politicians, even fictional characters are all persons of some kind that thousands or millions could form an emotional attachment with. This is part of why I think ensuring the maximum number of black elected officials while minimising racial tension (by having districts elect multiple seats so neither whites or blacks are stuck as local minorities with no chance of electing anyone) is so important; If a black politician finds success and gets his name attached to legislation that benefits poor whites, neo-Confederate rhetoric is going to wear thin for a lot of people. Even them just being living contradictions to decades of Antebellum propaganda of blacks being unintelligent, dangerous animals would force racists to make serious concessions in their worldview to not be taken as completely ridiculous.