I think the problem with your premise is that you're assuming everything had to do with slavery in that period.
Off the bat, the winners of the 1840 and 1848 Presidential Elections died in office during their first terms, so they were obviously precluded from winning two terms.
Martin Van Buren who won in 1836, but lost in 1840, was did in by the Panic of 1837, which was an economic contraction that was still lingering by the 1840 election.
Martin Van Buren and the American Political System Donald B. Cole
First one-term President you could directly relates to the issue of slavery relates to the 1844 Election of James K. Polk and his one-term pledge. Here, certainly, the issues surrounding slavery were causing federal politics to be less stable.
In 1844, the incumbent President, Tyler, who was unelected, submitted a Treaty to annex the state of Texas. This was an explosive issue that upended both parties since Texas would be admitted as a slave state and upset the sectional balance. Keep in mind that at this point, both parties had pro and anti-slavery factions.
Van Buren, who was originally the frontrunner for the Democratic nomination in 1844 ended up losing most his Southern support because he opposed immediate annexation of Texas, which deeply alienated slaveholders.
Polk, the eventual winner of the 1844 election, emerged as a compromise candidate, and to unite the Democratic Party that was divided by slavery, (as were the Whigs), pledged from the outset to be a one-term President in a bid to unite the party factions, precluding him from being a two-term President. And this will be very important for the remainder of the story, the Democrats agreed that in future conventions, the nominee must have the support of 2/3 of delegates, not a simple majority which had been the rule up to this point. * America 1844: Religious Fervor, Westward Expansion, and the Presidential Election That Transformed the Nation* John Bicknell
Polk: The Man Who Transformed the Presidency and America Walter R. Borneman
Franklin Pierce, who won the Presidency in 1852, and even failed to be renominated by the Democratic Party in 1856 can be also chalked up to the instability surrounding slavery. While President, Pierce signed the Kansas-Nebraska Act, which would allow slavery in certain territories if the majority of inhabitants supported it, violating the Missouri Compromise of 1820. This was deeply unpopular in the North.
Pierce ended up finishing 2nd place on the first round of balloting for the Democratic Nomination in 1856, (and remember the nominee needed 2/3 to win). Pierce ended up losing the nomination because he was seen as a polarizing figure on slavery and could not build a 2/3 majority. Not surprisingly, he lost to James Buchanan, who had been overseas the last 4 years and was able to avoid taking a position on these divisive issues relating to slavery.
The Constitutional and Political History of the United States. 1856-1859. Buchanan's Election - End of 35th Congress . Vol. VI Dr. H. von Holst
The winner in the 1856 election, James Buchanan was the oldest President ever elected at the time. He, like Polk, made a pledge to serve one-term from the outset. I've read too many competing explanations on why he did it, and none of them seem authoritative, some say it relates to his age, and others think he was copying Polk who tried to unite factions of the party divided over slavery.
/u/meteorblade beat me to the punch and has the essentials of what I was going to write covered, so I'll just add a few things.
Obviously deaths in office and one-term pledges aren't much to do with slavery, but the era between Jackson and Lincoln is one where slavery becomes an increasingly obvious and difficult issue. It was always important, but the national parties lost their limited ability to contain antislavery (and to a much lesser extent, proslavery) politics over the course of the era. Texas wanted annexation immediately on independence, but Jackson and Van Buren both wanted clear of it (Jackson refusing recognition of the Lone Star Republic until his last day in office) because there was no way it wouldn't be a huge win for slavery. Anglo Texas was built on slavery. Southerners went there and fought in its revolution to secure a slaveholding polity. So they go hands-off on the whole affair.
That angers plenty of Southerners and Southerners are the core of the Democratic coalition. The Whigs have a southern contingent too, but it's not generally as strong as the Democracy's. Consequently, the Whigs have a stronger antislavery wing. So Texas evolves into a Democratic litmus test, which in turn inclines the already leery Whigs further against it. The issue morphs into one with a clear partisan dimension, but also an important sectional sting. By the time Texas is annexed, it's a solidly sectional affair consummated by dodgy means (not a proper treaty) and amid great controversy.
Texas leads fairly immediately to the Mexican War, courtesy of James K. Polk's proslavery ambitions. The fallout from that is almost perfectly sectional rather than partisan. The initial outline for a settlement of slavery in the Mexican Cession is drafted by the Whiggiest of all Whigs, Henry Clay, and his plan completely falls through in the face of opposition on his own left and the more proslavery right. Those factions aren't entirely sectional -there's always proslavery Yankees- but antislavery Democrats are right there with the antislavery Whigs on the issue. The cobbled-together "compromise" is one that almost no one endorses, basically a set of separate bills that pass with squeaky margins by peeling off a few dissenters from each sectional consensus.
That goes over as well as could be expected, to the point where Whiggery implodes in the South from its inability to deliver a big win for slavery and its inability to muzzle antislavery Yankees. That leaves the party largely in the hands of antislavery northerners of one stripe or another. At the same time the Democracy takes similar, but less severe, blows in the North which facilitate a further transition the other way. It's always going to be difficult to pick out just when slavery became the thing -I would argue it had done so by 1800 and we should really look more at the rise of antislavery- but you can date its central, overt, and fairly consistent prominence in national affairs to about 1846-50.
So we are basically looking at the slavery question (think of it as "how much more can we have and how fast?" rather than "will we have it at all" though) blowing up national politics in a way it had only done intermittently before. But it's also true that the 1850s are not known for their crop of good presidents. That doesn't mean they necessarily lacked for strong leadership in the political sphere, though. It takes strong leadership to force through a massively controversial bill like the Kansas-Nebraska Act, which many northern Dems knew they would pay a heavy price for at the polls. Jefferson Davis most likely asserted that leadership through Franklin Pierce since he appears to have had a prime ministerial role for slavery policy in the Pierce administration. In that case, the presence of strong leadership ability and party discipline was far more long-term damaging than its lack.
Hi folks,
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