It goes back sixty years. Paul Bryant detested Tennessee from his playing days at UA and coaching days at Kentucky and Alabama. Tennessee was always tough to beat and Bob Neyland was considered best coach in the South in the 30s, 40s and 50s. Bama's athletic trainer, Jim Goosetree, a Tennessee grad, handed out cigars after a particularly tough win over Tennessee and the tradition took hold. Link: https://www.wvua23.com/its-tennessee-week-and-that-means-cigars-are-flying-off-the-shelves/#:~:text=It's tradition that Alabama fans,the beginning of a tradition.
My father played end at Alabama from 1930-32. In the Tennessee game of 1932, played in Birmingham under a constant downpour, Beatty Feathers of Tennessee and Johnny Cain of Alabama had a punting duel for the ages, over 30 punts. The coaches were calling punts on third down in case the punter bobbled a snap. Bama led 3-0 with a few minutes left but was backed up inside its own ten yard line. The field was all mud by then. I believe Darrell Royal, Mack Brown and even Nick Saban would have given Tennessee a safety and free punted it from the 20. However, Bama punted. Cain had averaged 48 yards per kick that day, but this one went off the side of his foot, badly. Tennessee had the ball inside Bama's 20 with very little time left. The Vols ran it. ran it and ran it again. Feathers finally ran it in for a TD and Tennessee won 7-3. God dammit.