Southern recomendations near I-40 (NM, TX, OK, AR or TN)

On the way down to 40, you might consider spending a few days in northern New Mexico. Santa Fe and Taos are both loaded with atmosphere, interesting plazas, great food, museums - linked by the "High Road" with old mission churches.

Around Albuquerque, you could visit Bandelier and Petroglyph National Monuments for some hiking. And even though the Balloon Fiesta is over, you could book a balloon ride over Albuquerque as a grand finale of your trip.