I think it would have been very contingent on exactly how the war and its aftermath went, but the two most obvious cleavages imo would be former Whigs and former Democrats on the one hand, and pro- and anti-administration on the other.
Assuming a Confederate victory in the Civil War, I could imagine a sort of neo-Jacksonian coalition forming around Davis's policies in support of a relatively centralized national government, a strong defense of slavery, glorification of the Confederate war effort, and expansionist foreign policy. On the other side you've got some of the old-time Whig aristocracy, Appalachian yeomen, urban merchants and factory owners, and anyone else who doesn't fit into the dominant Davis coalition (and isn't completely incompatible with the anti-Davis group).
ETA: although it's pretty closely related to the Democratic-Whig divide, another major potential cleavage worth mentioning might be the Deep South and the Upper South, which you can see reasonably well in this
1860 map.