If you have no luck hitchhiking out, I suggest to check posts on Craigslist, for the rides from both Nashville, Knoxville or Asheville, as there are many ride offers from drivers going towards New Orleans, and passing through Birmingham.
West towards Tuscaloosa and New Orleans
It's not very easy hitchhiking out, as many western suburbs of Birmingham along the I20 are considered dangerous. I tried hitching out of McCalla (20 miles west), but no one pulled over for 2.5 hrs. My local friend recommended to reach Tuscaloosa first, and hitch further from there, as it's safer and more relaxed city. Jan 2018