Except for the mountainous section between Las Cruces and Phoenix, any section of the Southern Tier is fine for winter touring. Temps do occasionally dip to below freezing, and rain can be expected Fld thru East Texas, but most days will be 60-70 degrees, nights in the 40s-50's. MSN Weather is good for checking avg temp/ppt ranges in any given area.
The problem with the SW in winter is staying out of the higher elevations. You can check elevations with Google Earth and/or
www.Bikely.com.
Check out central Texas/
Hill Country(lots of climbing.) Moderate temps, little rain, great roads and scenery. You could do a run along the Texas Gulf Coast all the way to Brownsville and back(flat) with expectations of nice temps, but more rain chances. Or South of Phoenix and Tucson, including
Tombstone, (mostly moderate climbing.) Almost no chance of rain.
Map Link here.