Not necessarily. An evenly split senate can tilt one way or the other depending upon who is in the White House. Lincoln made it very clear he was willing to influence the direction of that tilt if necessary.
But the Senate was not evenly split. There were 38 Democrats and 26 Republicans and 2 from the American party. If the tariff was really the issue then the southerners had more than enought votes to kill it. So why didn't they?