There is another option for east of the segment you showed, so you can bypass the I-40 stretch between Memphis and Little Rock mentioned by 1Boyscout…
If you take I-40, get off I-40 onto US-287 at Amarillo, TX.
Stay on US-287 until you intersect US-82 at Henrietta, TX. This will take you around Dallas/Fort Worth.
Stay on US-82 until you intersect US-69 at Bells, TX.
Stay on US-69 until you hit I-20 between Lindale, TX and Tyler, TX.
Follow I-20 until you intersect I-459 at McCalla, AL.
Loop around Birmingham on I-459 until you rejoin I-20, all the way to South Carolina.
The only bad part of this route is passing through Atlanta, where traffic is a nightmare. But looping around Atlanta is even worse.
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