My preference, same start, but take I-530/US-65 out of Little Rock. It is US-65 you are following. I usually turn east on US-82 at Lake Village, cross into Mississippi at Greenville. Then east to US-49W, follow that South to I-10. I like this route for smooth traffic flow, and convenience of potential gas and rest stops.
Alternatives I've used include US-65 all the way to I-20, US-61 down from Greenville to I-20, both cases headed to Jackson and US-49. But US-65 in Louisiana and 61 south from Greenville are slower and rougher road, and I-20 between Talulah and Jackson is some of the roughest Interstate I've ever driven.
Since you are not going to Gulfport, you could leave US-49 south of Hattiesburg, follow US-98 into Mobile. I have no recent experience with this route, last used it about 10 years ago.
I take these routes to get away from the heavy, high speed truck traffic on I-40, I-55.
For a real leisurely drive, you can follow US-98 out of Mobile, see the Alabama Gulf Coast. I usually wait until I am past Pensacola before dropping onto the coastal route.