OK, here is a variation from the direct route across Arkansas and Oklahoma from Memphis, which follows I-40.
In Little Rock, instead of staying on I-40, you could take the south bypass to I-30 to Texarkana. You could then follow US 82 all the way across north Texas to US 287 near Wichita Falls. US 287 to Amarillo where you would rejoin I-40 to Clines Corners and up to Santa Fe.
I must admit that this is basically the route we follow from Paris, Texas to Santa Fe.
Of course, I think it would be worth planning a stayover in East Texas.