As Thomas B. Alexander writes in "Persistent Whiggery in the Confederate South, 1860-1877":

"Students of the South's political history are not unaware that the Constitutional Union party of 1860 was substantially a continuation of the Whig party of the South..."

It's no surprise, then, that support for the Constitutional Unionists in 1860 was strongly, but not entirely, correlated to support for the Whig Party back in 1848 when the Whigs won, and more especially in 1852 when the Whigs lost, just as the Constitutional Unionists did eight years later.

A stand-in answer to this question can therefore be found in the book The Rise and Fall of the American Whig Party by Michael Holt. On pages 116-118 of that book, the author identifies several regions of the South that tended to support the Whigs, which largely overlap with Constitutional Unionist support. The gist of his argument is:

Did the region grow tobacco or cotton? Or did the region grow sugar, hemp, or other crops? The slaveholding regions that grew tobacco and cotton tended to largely support the Democrats. The regions that didn't tended to support the Whigs. This is because the protectionist tariffs championed by the Whigs made exports more expensive, which was seen to drive down demand, and tobacco and cotton growers were more reliant on exports. On the flipside, the tariffs propped up the price of commodities aimed at the domestic market, by making foreign-grown imports more expensive. This was beneficial to growers of sugar, hemp, and several other crops, so they tended to support the Whigs. So did their community at large, whose jobs and economic well-being depended on the profitability of those crops. Thus, you see places like Kentucky, Tennessee, and parts of the Lower South break toward the Whigs, where cotton and tobacco weren't the main cash crops.

Another issue: did the area have a notable Catholic or white immigrant minority community? If so, then the majority Protestant and native-born community tended to vote Whig, because they were historically the anti-immigration party, whereas the Democrats courted the immigrant and Catholic vote. Thus, parts of Louisiana and, generally, port cities/towns adjacent to the Mississippi River with communities of immigrant dockworkers tended to vote for the Whigs over the Democrats.

Holt says that the exceptions to these rules are mostly found in Virginia and North Carolina, where some cotton- and tobacco-producing counties voted Whig, because they supported the Whigs' "internal improvements" infrastructure programs that proposed the federal government build schools, post offices, bridges, dams, etc., and saw this as more valuable than the effect that the Whigs' protectionist tariffs had on their crops. This may also partially explain the support in areas adjacent to the Mississippi River--dams and bridges and their upkeep could drastically improve living and economic conditions in those areas.

Partisan identification doesn't simply go away en masse because one party or another supports a specific policy, so even though the Southern Democrats were championing "states' rights" to own slaves in 1860, former Whigs who were slaveholders weren't always quick to join up with the Democrats' cause (although some did, like Robert E. Lee, a former Whig who voted for Breckinridge). The 1856 county map shows that Know-Nothing candidate and former Whig and U.S. President Millard Fillmore ran strongest in the Whig strongholds of 1848 and 1852, and it was these same counties where Constitutional Unionist John Bell mostly ran the strongest in 1860.

While the Constitutional Unionists' platform wasn't anything more than to preserve the Union, as was Bell's acceptance speech, this was because they were trying to appeal to both Democrats and former Whigs (as well as conservative Republicans in the North and West) and did not want to alienate any portion of their target electorate by being seen to prefer one party's historic policies over the other. Nevertheless, a considerable majority of the Constitutional Unionists had come from the Whig party, and not from disaffected Southern Democrats. Most Democrats voted enthusiastically for Breckinridge--or else, if not sold on Breckinridge, then Douglas.

John Bell, after all, had been affiliated with the Southern Whigs since 1835, and had only left the party when there was no party left. Most of the other candidates proposed at the 1860 Constitutional Unionist convention--John J. Crittenden, Edward Everett, William A. Graham, and William Rives--had been aligned with the Whigs since the 1830s or 40s. The only exception was Sam Houston of Texas, who was nominally a Democrat, but a strong "Union Democrat".

As Holt says, the Constitutional Unionist Party actually had its roots in the 1852 election, when some Georgia Whigs almost nominated a separate candidate from Winfield Scott, considering him too divisive in the South. They ultimately fell in line, and were pleased enough with known quantity Millard Fillmore in 1856 that another Constitutional Unionist candidate wasn't proposed until 1860, when the Know-Nothings had fallen apart and Breckinridge was unacceptable to the former Whig faithful.

Thus, the fact that fragments of the former Whig coalition in the South voted for the Constitutional Unionists isn't really any surprise. Just as with any era, people aren't all of one mind, and slaveholders and their communities weren't any exception. The Southern Democrats were surely the more vocal pro-slavery side of politics, but not every slaveholder saw their policies as beneficial to slavery. Many saw the extremist policies of the Democrats, largely predicted to lead to war, to be worse, not better, for the long-term survival of slavery. A loss in war could end slavery, but the Constitutional protections of slavery could sustain it for the foreseeable future.