OTL the lower class whites were really screwed by the elites. Social and economic mobility was dramatically less than in the former CSA states than elsewhere in the USA. Public education for whites, especially outside of urban areas, was terrible and any sort of infrastructure that was state financed and controlled (the bulk by far until the New Deal) was pretty pathetic, decent roads served the needs of the elites and once you got off those terrible was a good day. Cronyism, nepotism, and corruption was endemic. Why did the lower class/poor whites not demand better - they had the reality that the lowest white was forever above the highest black, that whatever improvements such as roads, sewers, running water, schools, etc that did come about would go to them and pretty much only them. While acceptance of crap governance was not universal, the Jim Crow system satisfied the bulk of those not in the power structure, and the reality was while rising was difficult it was not impossible for the white man, but forever denied to the black man.
In an independent CSA you have the reinforcement of the victory flush, plus the 2/3-3/4 of the population that did not own slaves not only had the white over black situation of Jim Crow but could aspire to become slave owners. IMHO you would actually see slave ownership become MORE distributed than in 1860 for several reasons. You were beginning to see a "slave surplus" as the number of slaves in the upper south decreased due to economic reasons, changes in agricultural products etc. As you get some levels of mechanization, and with the better industrial jobs reserved for whites (and there will be some increase in industrialization, just not a lot), plus natural increase, the price of slaves as a commodity will decrease meaning owning a small number of slaves becomes economic for more folks. Much like affording even a small number of domestics became a symbol of the middle class (not the huge staffs of the very wealthy) you will see shop owners and better off yeoman farmers owning a domestic slave or three, or slave low level shop assistants, or a farm worker or two. Every additional citizen who graduates to the slave owning class becomes another strong adherent of keeping the system. For the married on this board just imagine the impact of wives complaining to husbands "if we have to pay our cook and maid, give them days off, we simply won't be able to afford them - and how will I ever manage this household without them...". I doubt THAT husband will vote for emancipation.