I've lived in Kansas and Wyoming and have biked the Great Plains from the Dakotas to Texas.
While what some others here have said about winds in generally true, it lacks in the specifics. First, yes, winds in the Plains are generally either northerly or southerly, but they do vary by season and location. Second, winds are lighter in the eastern plains - where there is a bit more tree cover - and stronger the further west you go - where trees become exceedingly rare. Third, southerly winds predominate in the summer with a switchover to northerly during the fall. Fourth, winds are slightly southeasterly in the eastern plains but become more pronounced southwesterly in the western plains.
If you want to check out wind roses, the most appropriate for your proposed trip are those from Amarillo, TX and Dodge City, KS. Or course, these are only two stations - - winds vary, even in the Plains, according to local topography. Note that Amarillo has a clear southwesterly vector in Sept, Oct & Nov. The westerly component comes a bit later in Dodge City.
Having also driven frequently between Wyoming and northern New Mexico, I can attest that frequent, strong southwesterly winds are common in the stretch of I-25 between Santa Fe and Raton. In dozens of trips, I can easily remember either bucking the wind southbound or sailing on it northbound - - driving - - in all seasons. I would have hated life on a bicycle heading southwest.
Now, I am just one person, but I believe that if you were to ask people living in the High Plains - such as in Clayton, NM or Guymon, OK or La Junta, CO or Pampa, TX - you would find general agreement on southwesterly winds.
In sum, in the part of your proposed trip with the strongest winds and least cover, in the autumn the prevailing direction is southwesterly. YMMV.