Just looking at the map, one possibility would be to take US 412 to Jackson, TN. I am not familiar with this part of the route so cannot make any specific recommendation, but it looks reasonable on the map.
From Jackson you could take US 45 all the way to Meridian, MS. Again, I have to say I have not traveled on this route.
At Meridian you could take HY 19 which becomes HY 10 in Alabama to Butler, AL then pick up HY 17 at Butler all the way down to US 84 then over to Coffeeville.
Now, I have driven this route from Meridian to Coffeeville. These are pretty narrow state roads, but I have towed a 34' fifth-wheel on them several times. We had friends who lived in Monroeville, AL and this is the way we often traveled.
Don't take this a advice or recommendation. It comes from a study of the map and your preference to avoid Memphis, plus a familiarity with the last part of the route.