We can almost immediately cross Alabama, Mississippi, and Tennessee off the list.
Though Alabama elected a Democratic Senator about two-and-a-half years ago and Tennessee's native son drew within four points of George W. Bush in the 2000 Presidential Election, these states are just too far inelastic to become competitive for the blues within the near future (at the Presidential level, anyway).
My pick would be South Carolina, but it would be a personal toss-up between that and Louisiana.
In order for any Democrat to make any of these states competitive, they would have to run a charismatic southerner with JBE-type politics against a weak (and potentially scandalized) Republican opponent. And even then, I'm not sure that would be enough to outright win.
Mississippi has hyper-racialized partisan voting habits (moreso than SC or LA, probably) but also a larger African American electorate than any other state. MS is on-track to being plurality Black as early as 2040. That alone makes its the most likely state in the Deep South to eventually become competitive.
I'll defer to you as you're obviously much more in-tune with your state, Del Tachi 
I would say it becomes Dem by 2032 as Republicans wont be able to get the numbers they need with White voters to keep that state R. I think they need 80% to win the state but with millennials and gen z proportion of the population growing that wont happen by 2032 so the state would go D.
Also given how small it is the GOP wont do anything to save it while for GA and definitely TX the GOP will given how it will be hard for them to win without GA and without TX well then the GOP is done even if trends makes MN MI PA WI NH ME CT DE all Republican. So GOP will adapt to save TX but for MS they wont as parties rarely adapt to save states that are worth less than 10 EV
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