First, secede, not succeed.
Second, this is a difficult question to answer. Slavery was virtually non-existent in the western territories even before Lincoln took office. It was flourishing in the Deep South and probably would have continued to do so for several years had the War not happened. The story would probably have been different in the the Upper South. Remember, that region may not have seceded had Lincoln not called for an invading army to put down the “rebellion,” and had rejected it outright before April 1861. Thus, slavery would probably have died out in VA, NC, MD, DE, KY, AR, and TN. It would have taken some time, but I don’t think it would have survived past the 1880s.