Interstate 10 pretty much runs the Southern route all the way out and we took this in October because we were concerned about driving through the mountains with the possibility of snow.
We started in Minnesota and Iowa had already received snow so we straight-lined down I-35. We did zip up to I-40 to do Oklahoma City, Amarillo and Albuquerque then dropped down I-25 to do Las Cruces and jumping back on I-10.
Roads were very good and lots of good campgrounds close to the route.
Scenery from Las Cruces to LA is not as good as I-40 but safer with winter weather.