first you need to explain away why what happened OTL wouldn't happen in your ATL, since the POD is after OTL civil war. The events that lead to the OTL not happening are going to have an effect on what comes after.
It's pretty basic: you need the south to feel that the moment is at hand where their way of life is threatened to the point that not acting is accepting the south's demise. They obviously felt that point was at Lincoln's election. If you wish that away, the next point comes when the situation gets even more dire. If you want a point, just imagine a northern abolitionist introducing legislation to abolish slavery. The south was obviously getting fed up with northern intrusion, so it's not that tough. What makes it tough is somehow keeping the OTL reason the war started, while ATL stopping the war from happening.