You might want to look up the Morrill Tariff.
By that point seven states had already seceded and presumably their senators and representatives had already left Washington and didn't vote. The Confederate Congress had already been formed and Jefferson Davis had already been sworn in. So I doubt people in those states were that mad about tariffs in a country they'd already decided to leave.
It should be noted that the tariff passed in 1861 was lower than the tariff passed later to pay for the war. It was generally recognized that the tariff -- which went down when the Democrats took over -- would go back up when the Democrats left power, but Southerners in Congress would have had an important role in determining how much tariffs would change, if they decided to use their influence in Congress, rather than just give up on the country.