Well the OP doesn't say how the CSA gains it independence, whether there is a war, or whether the election of Lincoln was the catalyst for secession.
As such, what if the Republican party is more disorganized and factured in 1859-1860 and that the Democrats don't split in 1860, such that the Democratic candidate, say its Douglas wins the presidential election over the Republican, Lincoln or someone else. Nevertheless, the Republicans have made significant gains and hold majorities in both houses of Congress.
The Southern States see the writing on the wall and petition/sue for secession. President Douglas puts up a bit of resistance, but in the end, pretty much lets the 7 states of the deep south go and form CSA. Basically agreements are made to pay the federal government for federal property, etc.
Needless to say the more radical Republicans go nuts, and many more moderate republicans become more radical. With the senators and representatives of the CSA states now gone, Douglas has absolutely zero support in Congress. He is soon impeached and removed from office.
However, by this point, war with the CSA can't get any traction, however, the radical abolitionists and others opposed to slavery feel betrayed by the federal government and as such their numbers grow, they become better organized and funded, and with the federal government turning a blind eye, begin campaigns to incite uprisings and rebellions, arm slaves, etc., in the CSA. Getting CSA slaves to freedom becomes much easier as they only need to get to Ohio, Pennsylvania, etc., and not Canada. The slave states that remained in the Union will have become marginalized with very little power to try to keep fugitive slave laws in force. But will those states try to go and join their brothers in the CSA?