I would leave Jax Friday afternoon/evening and drive past Atlanta that night. It's 500 miles from there to St Lou and stay at the CG recommended above. I usually try and stop on the far side of any large town but on Sunday morning traffic should be OK for the two day trip to Rushmore (1000 miles). I would go through Cody into Ystone's east entrance... 440 miles. This gets you to Ystone on Tuesday.
I would plan on a day and a half in the Tetons staying at Colter Bay. This lets you take 287 without backtracking to Rawlins and then I-80 to Lincoln NB then KC and St Lou. 2200 miles home in four days. Good luck.