You might be surprised by what's possible with enough propaganda and panic. Some state conventions rejected secession, then voted for it. At least one state was supposed to have a referendum on secession and didn't.
I suppose support for secession in most of the Confederate states was stronger than support for remaining with the union, but tactics and timing had a lot to do with shaping the public mood and making events happen.
“Some state conventions rejected secession, then voted for it.”
So what? A vote requires a majority to pass. You can’t claim a “small minority” passed secession when they got a majority to vote for it. That’s inane.
“At least one state was supposed to have a referendum on secession and didn’t.”
“Supposed to” according to who? Secession was not a legally defined process, so there was no precedent about how it was “supposed to” happen, save perhaps the American Revolution, which was initiated by legislatures and not referendums.