Perhaps less than you might think. There is a new paper by Mario L. Chacon and Jeff I. Jensen:
We use the Southern secession movement of 1860-1861 to study how elites in democracy enact their preferred policies. Most states used specially convened conventions to determine whether or not to secede from the Union. We argue that although the delegates of these conventions were popularly elected, the electoral rules favored slaveholders. Using an original dataset of representation in each convention, we first demonstrate that slave-intensive districts were systematically overrepresented. Slaveholders were also spatially concentrated and could thereby obtain local pluralities in favor of secession more easily. As a result of these electoral biases, less than 10% of the electorate was sufficient to elect a majority of delegates in four of the six original Confederate states. We also show how delegates representing slave-intensive counties were more likely to support secession. These factors explain the disproportionate influence of slaveholders during the crisis and why secessionists strategically chose conventions over statewide referenda.
Not entirely unlike the first American secession!
For the pointer I thank the excellent Kevin Lewis.