Well, to stop the decline of black percentages in the South, you'd need the following.
No Great Migration: The black population, as a percentage, mainly declined pre-1950 due to migration to cities in the North for greater economic opportunity. A number of factors played into this - a desire to escape Jim Crow laws, the boll weevil infestation in the 1910s, the abrupt end of mass immigration in 1924 requiring a new labor force in much of the North, and the expansion of wartime defense industries just as the number of able-bodied men declined. Essentially, you'd need to dramatically increase economic opportunity within the South, so that even when black families moved off the farm, they moved into cities like Birmigham and New Orleans to work in factories, not Chicago and Detroit.
No Sun Belt: Post 1950, not only were blacks moving out of the South, whites began moving into many regions. You'd need to stop thriving metropolises from forming like Miami, Houston, and Atlanta, instead having smaller cities which only draw on a regional, more black population. The mass settlement of the South was made possible by the development of air conditioning, which is hard to butterfly away. However, the other major reasons for the formation of the sun belt were political. For example, businesses relocated to the south because it was cheaper due to "right-to-work" laws which made operating a union difficult, along with the general anti-union attitude of most white workers outside of the coal mines. This could easily have worked out differently if it wasn't for the historical accident of Taft-Hartley.
These, particularly the first, would require PODs dating all the way back to a successful Reconstruction, as a southern, black class of capitalists would probably be a necessity. It's a difficult tension however, as a successful enough reconstruction would presumably lessen racism in the North as well, meaning blacks would move wherever they pleased, and no one would mind much.
More importantly, at best these changes would hold the racial balance to within 1860 norms, or nudge it upward just a bit if blacks have larger families. To really up the percentage, I think you'd either need whites to decline, or blacks to migrate in from somewhere else. The former is unlikely unless radical reconstructionists were really out for dixie blood - even massive land confiscations would lead to mainly wealthy families being effected. As to blacks migrating in, it could just barely be possible that they migrate in from the Caribbean or something, but that would necessitate national, state, and local governments which wanted millions of black immigrants, along with a local black capitalist class to employ them.
I think it's much, much easier to get an overwhelmingly black deep south though. Let's say land redistribution becomes common in SC, GA, FL, AL, MS, and LA, but is largely restricted to those states. There is a small outflow of the former white gentry, who having lost their property migrate into the upper south, west, or out of the country altogether. Some black farmers become wealthy, and by the 1880s form the basis of a black upper class, who uses their capital to get into the manufacturing sector. Just like white factory owners often preferred to hire their own before anti-discrimination laws, so to black factory owners. Initially they only pull from local rural areas, but as time passes, more and more blacks from the upper south move further southward. Soon there is a demand for immigrant labor as there was elsewhere in the country, however in part in the deep south it is taken up by Afro-Caribbean peoples like Haitians, who were in some cases directly sponsored to migrate by the well-to-do and well-connected in the deep south. In many cases the immigrants weren't black, coming from Cuba, Central America, or even Europe. However, within a few generations, they acculturate and often intermarry into the black population. Indeed, as blacks rose in economic clout through the South, some poor native-born whites (women in particular) married into black families.
Okay, so this seems an ASB scenario, as it basically requires a nation within a nation. It seems unlikely to me that local black capitalists could really compete with Northern magnates, for example. But the only real alternative I can think of would be a severe form of Malaria, like Plasmodium falciparum, sweeping through the South and killing off much of the white population, remaining endemic in the southern lowlands until DDT or an analogue removed it from the South in the 1950s.