I recently have been researching my ancestors in East Tennessee, and read "War at Every Door" and "Mountain Rebels." One of these books had tables showing the county by county votes. It was quite telling. In the June 1861 vote for secession, as I recall, in Cocke County (where my East Tennessee ancestors lived) secession was voted down by about a two to one margin. However, in Fayette County (where another pair of my ggg grandarents lived in 1860) secession was approved by over 97%. Both are good books; I will check which one had the charts.
There were two votes. Governor Isham Harris was pro secession, and requested a referendum on whether Tennessee should hold a convention to consider secession. As I recall, you voted yes or no for the convention, and for the delegate in case the convention vote passed. The referendum was held February 9, 1861 and the unionist position won (i.e., the position not to have a convention).
After shots were fired at Sumter in April, and Lincoln called for 75,000 volunteers to put down the rebellion, attitudes changed in Tennessee (as well as N. Carolina, Arkansas, and Virginia, and Missouri, although it did not eventually seceed). Persons who had voted Unionist in February now felt there was no choice but to stand by the other southern states. In May, the Legislature withdrew the state from the Union and set another referendum, to approve of secession, for June. In the June referendum, Secession was approved (although the majority of the eastern counties continued to vote against secession).