The actual trigger point wasn't a Senate Majority (which was given up in 1850), but rather a more general sense of increasing "containment" by the South. Early in the life of the country, the slave/free balance remained even because there was land suitable for both to expand.
1848 and the War with Mexico added some land, but this land had no slaves and there was a contingent in Congress that didn't want to extend Slavery - the Free Soilers. The Compromise of 1850 kept them from extending it to California, and the rest of the land out west seemed unsuitable for large scale slave importing, it being desert and waste for the most part.
Enter Kansas. By the Missouri Compromise of 1820, this was to be free soil, but the slave holding areas of Missouri were in the North, just adjacent to Kansas and didn't want free ground so close. These fellows felt surrounded, and the South tricked Douglas into allowing slavery in as "Popular Sovereignty". It just got worse when the real free soil settlers soon outnumbered the slave settlers and the Missourians who rode in to vote about 10-1, despite rigged elections and even a mini Civil War. Kansas was going to end up a free state regardless.
This political disaster elevated Free Soilers (and others) into a new major party, the Republicans, dedicated to locking slavery out of any new territory. The strength of this party in the North and the fact that many Northern Democratic voters weren't interested in new slave states either meant a compromise that might result in a new slave state was out of the question going forward.
In 1857, one last stab at 'escape' was made with the Dred Scott decision by the Supreme Court, which purported that Slavery could not be restricted anywhere, in the Territories nor in Free States. This absurd conclusion, which in essence tried to make the Republican Party unconstitutional and make all states slave states, assured the rupture of the Democratic Party and the win by Republicans in 1860.
At that point, those fighting containment could only do it outside the US. The radicals managed to pull some states out, and then induce a war that would get nearly all the slave states to follow. They were terrified that unless it came to arms, a compromise like those in 1832 or 1850 would follow.