Okay, so.
The thing you have to remember is that, for most of the antebellum period, the loony fringe camp in US politics were the abolitionists. There was a broad consensus between southern planters, northern Democrats and Cotton Whigs that slavery, if morally wrong, was necessary to maintain the economy, that if the cotton plantations were worked by free labour they'd get outcompeted by the British and the country would get driven to financial ruin. To this end, the House of Representatives enforced a gag rule that strictly forbade it from taking any action on any motion of any kind that had anything to do with the institution of slavery, and the altogether chummier Senate kept discussion out by gentlemen's agreement. Of course, this was never without controversy, and a few members of the House - notably former President John Quincy Adams, who brought petitions against slavery throughout his congressional career (from 1831 until his death in 1848) - continued to resist it. But those members, and the radical, often Evangelical Christian "movement abolitionists" in the North even more so, were always the beleaguered minority. They were the ones who had to take action to oppose slavery, the supporters of slavery never had to take action to justify their position.
Matt Karp's book This Vast Southern Empire, which I recommend if you're interested in reading deeper on this, goes into detail about how the southern planter interest was able to use this situation to further their aims in foreign policy - the US Navy basically exists in its current form because the planter class was terrified of the British coming to take their slaves from them, and were convinced the federal government needed to take action to defend the coast. The Mexican-American War was from the very beginning a project to extend US land holdings below the Missouri Compromise line, and by extension secure more land for plantation agriculture. When Texas seceded from Mexico, they wrote up what was essentially a boilerplate US state constitution, and the goal of their "revolution" was always to secure their land for the US and for slavery rather than to build any sort of new nation.
So the planter class of the 1840s would never have contemplated secession - why would they, when the federal government was working for their class interests and Congress was gagged to prevent this from ever changing? The only reason secession became a viable option was that this broad pro-slavery consensus (what would later be dubbed the "Slave Power") came crashing down over the 1850s as a result of the planter class pushing its demands in a way that offended opinion across the North. This probably starts around the Compromise of 1850, which essentially consisted of the South giving up its claim to southern California as a slave state in exchange for the North giving up its de facto nullification of federal fugitive slave laws. The institution of slavery had always been tempered somewhat in the eyes of the North by the fact that slaves who fled north were usually tacitly given their freedom, and that this meant the violence inherent to the slave economy was never felt north of the Mason-Dixon line. The new Fugitive Slave Act made the ramblings of the "movement abolitionists" that slavery would inevitably envelop the whole United States seem a lot less crazy, and when Kansas-Nebraska came along and basically made it impossible to keep slavery out of the territories, the fire was stoked yet again.
If you wanted to compress the course of events leading to secession into a text-chain format, it might look something like this:
- The entire US economy, North and South, is dependent on cheap cotton, which in turn is dependent on the maintenance of slavery to keep production costs down. The plantation economy, besides slavery, needs to constantly take over new land in order to maintain growth - by the 1840s, there's not much good cotton land left east of the Mississippi.
- The fact that the North needs cheap cotton means the cotton planter class (almost politically synonymous with "the South" at this point, because even if other southern voting blocs exist, only the planters have any real clout) has a massively disproportionate influence on federal politics.
- The fact that the planters need new land means they use this disproportionate influence to impose their will on the federal government and make it pursue a more and more radically pro-slavery foreign policy - build a navy, press claims on Texas, Cuba (that one failed, luckily) and California, make the North enforce fugitive slave laws, open all the territories to slavery.
- Popular opinion in the North turns against the planters for imposing their will on the federal government, causing more and more northerners to question the institution of slavery itself.
- The planters look at popular opinion in the North turning against them and starts to wonder if they're going to be able to maintain their disproportionate influence over the federal government - the North, after all, has a lot more people.
- The North takes advantage of the fact that it has a lot more people to elect more and more anti-slavery candidates to Congress, causing the congressional Slave Power to break down as of the 1858 midterms.
- The planters look at the congressional Slave Power breaking down and decide maybe this is it then. Politicians start openly calling for secession, and state militias start to ready their guns (aided by sympathetic figures inside the federal government).
- The North pools its electoral votes to elect Abraham Lincoln to the Presidency, his campaign promising to restrict slavery to the states where it exists - notably not actually calling for the abolition of slavery, but in practice, the plantation economy would suffocate if not allowed to expand.
- The planters look at Lincoln winning, say "fuck it, last straw" and secede.
So you can see why secession during the Mexican War wouldn't really make sense. At that point, the Slave Power was close to its zenith - the US was literally fighting a war on behalf of the slave economy, and planters were sending their sons to fight in Mexico by the droves while the North was at best apathetic about the whole thing.
As for secession before the Mexican War, or in a TL where it doesn't happen, that's different. The annexation of Texas was probably the closest call of any step in this whole process, and could easily go the other way. And if Congress says that actually, no, you can't just take an entire province from Mexico just because the carpetbagger "locals" say they want to be Americans, that throws a serious spanner in the works of the Slave Power. I'm still not sure if secession would be the way forward, because IOTL it only came about as an option once it was clear that the South wouldn't under any circumstances be able to stay in the Union and maintain the slave economy. It's possible that things start with no Texas and deteriorate from there - especially so if Congress decides to pull some chicanery to open new slave territory inside the US - but you'd probably have to go at least 5-10 years from the Texas annexation failing for things to get so bad that the South decides to cut its losses.
