If I remember my statistics correctly, at the time of the ACW approximately one in four southern households even owned one slave, the percentage with multiple slaves was much smaller. Most whites in the south had no financial stake in the continuation of slavery, however there was a huge social one. With blacks, even free blacks, being a permanent underclass where the highest black was lower then the lowest white, this was a plus for the elites. Furthermore any white, should they get the money, could aspire to become a slave owner. All of this worked in the favor of the southern elites who saw themselves in a vision of a squireocracy with distinct and difficult to cross class lines. There was talk in the CSA political circles of, after a victory, of the reimposition of property qualifications for the franchise, but the permanent presence of slaves (and free blacks were going to go away) would keep the whites who did not qualify for franchise satisfied.
Many of the southerners were all about preserving their way of "civilized" life against the northern money grubbers, parvenus, etc. The more perceptive ones realized that even with the 3/5 rule and the northern democrats backing the south up, that sheer numbers in the House of Representatives were turning against them rapidly and should the differences between northern and southern democrats diverge this would be huge. The other reality was that while you might get some new states to pass slave constitutions that was highly unlikely, and even if passed they would probably be nullified in short order meaning the Senate was going to go further against them. The really sharp ones, if there were any, realized that the financial balance where the south's agricultural products were the big money maker in the USA was shifting rapidly, and norther industry as well as commercial agriculture (wheat for example) was going to be the major money maker.
Put all that together, and the "slave power" or disproportionate political power the south had was going to fade, which meant that slavery was going to be in trouble - liberty laws in the north, economic issues, and so forth. Hence the attempts to expand in places perceived as being more slave amenable, and eventually secession.