I would go US-331 to I-65 (or go a little west further south to get on 65 sooner) to I-459 to I-59 to I-24 into Chattanooga. I just did a similar route a little farther west from Navarre.
Larry, Beverly & Pat
Pups - Romeo, Teddy Bear, Frankie
2013 Thor Tuscany, 45', Freightliner chassis, Cummins ISL10, towing a Buick Enclave