A common belief I see on the board is that, in a CS Victory TL, there will be another war between the Confederacy and the Union sooner or later. I thought a thread on this particular question would make for some good discussion.
I myself do not adhere to this belief. Barring foreign intervention, the only way the South can win the war is for Northern political and social will to continue the war to collapse, resulting in the election of a peace party in 1864. This obviously means that the Northern people are not going to be anxious for a resumption in fighting in the immediate aftermath of the war. By the time the Republicans are once again in office, years will have gone by, during which time the Union and Confederacy will have begun to get used to one another as independent nations.
The Confederacy is going to be in very bad shape in the aftermath of the war, heavily in debt and with its infrastructure badly damaged. I think it could be a common thought in Northern minds that it's not worth the trouble to go to war against to bring such a ruined region back into the Union. More than anything else, I think the average Northern voter would have a "Good Riddance!" attitude towards the Confederacy.
I also don't see the strategic interests of the Union and the Confederacy being much in conflict unless the Confederacy embarks on an expansionist foreign policy, which they abjectly lack the strength to do. There is absolutely no reason for the South to block Union access to the Mississippi River, since it is so obviously in their own economic interest to allow Union shipping to use the river as much as it wants.