North: Democrats, Socialists, and later Progressives, probably in that order
South: Democrats, Whigs, and maybe a Christian Center party
Otherwise it might also depend on how and with what territory the Confederacy leaves. Maybe Oklahoma/Indian Territory has a Native American party?
The CSA pretty much has Oklahoma in the bag, it remained largely in Confederate control (outside Union-held chunks of the Cherokee nation, which was pro-Confederate from the onset of the war) until nearly 2 months after Lee surrendered.
The Two-Party state in the Confederacy is going to evolve slowly, but the rift is right there in the beginning.
The old hardliner fire-eaters, nationalists and conservatives form a Democratic Party that pretty much is the decendent of the pre-War Southern Democrats. This, after the "War of Secession" is going to be made up of the pro-Davis faction or if you will the of Confederate politicians during the war. The likes of these include Robert E. Lee and Braxton Bragg. This party is going to be rabidly State's Rights, decentralization to it's fullest, especially in strongholds like South Carolina and Mississippi.
The new Party, will be made of largely anti-Davis elements in the Confederate government and military. This party will be largely just as Conservative as the Democrats, but with at least 2 wings, one more in line with the Democrats, and one with a slight and gradual moderate-liberal slant. It will include elements like ex-Unionists in East Tennessee (and Kentucky, if it's in the CSA), populists, former Whigs and former Know-Nothings! Those in this group will include James Longstreet, Joseph E. Johnston, PGT Beauregard, and Alexander Stephens. This party's stance will focus heavily on maintaining the Confederate Constitution, but at the same time pushing a little on a few slight centralized ideas in the Confederate government. The names of this party here on AH are robertp6165's "Liberty Party" in his
Black and the Gray and Turquoise Blue's "Constitution Party" in
Two Americas are ones I like.