Outright Unionists somewhat outnumbered outright pro-Confederates in Kentucky during secession winter, but most Kentuckians were neutral fence-sitters. Lincoln's adroit political maneuvering was enough IOTL to keep Kentucky in the Union, but had he made a few missteps it might have been enough to push Kentucky into secession.
If the Fort Sumter crisis had gone a different way and the end result was to have the appearance of Northern aggression against the South instead of the other way around, Kentucky might have been pushed into secession. Remember that, even IOTL, Kentucky refused to contribute the requested regiments to the United States as part of the initial call up issued by Lincoln and tried to remain neutral as long as possible. So if Lincoln had mishandled the initial crisis and came out looking like the aggressor, the pro-secession elements might not have boycotted the 1861 elections and the state legislature would thus have had a much stronger secessionist element (though likely not a majority). This, combined with inevitably movements of Union forces into Kentucky, might have led to a Kentucky secession, unless Confederate troops move in first, as they did IOTL.
Alternatively, one could envision a scenario in which General Polk does not move into Columbus in September, 1861, as he did IOTL, and the Union forces occupy Paducah, thereby being the first to commit a major violation of Kentucky neutrality. Combined with the political fallout from John Fremont unilateral emancipation proclamation, might have been enough to drive the fence-sitters into the pro-Confederate camp. The legal government would still be Unionists, since this POD would have taken place after the state elections in August, but many more Kentuckians would have flocked to the Confederate Kentucky government than was the case IOTL and it would thereby have had greater credibility with the people.