While there are no maps of the "old Southwest" before colonization that we know of, the French and Spanish governments left behind some wonderful cartography. This 1783 map, published in Paris, is one such example.
This is considered a rare map because it was one of the first to label the United States, which was founded in the Peace of Paris in 1783. Of course, that labeling was in French... see the full map here: https://www.raremaps.com/gallery/detail/81052/carte-du-mexique-et-des-etats-unis-damerique-partie-meridi-dezauche.
What I really like about this map is its combining of European claims and native America. It shows the extent of what New Mexico/New Spain claimed, which in 1783 included, roughly, all of today's Louisiana and Texas, and much of Arkansas and Oklahoma. While the European political information stemmed from the 1783 maps used at the peace conference between the United States and Britain, the villages and nations of the original people along the Red River - labeled the R. Rouge ou R. De Marne - comes from French maps in the first half of the 18th century.
The Cadodaquoi - the Caddo Nation - is prominently labeled at Natchitoches, and north along the Red River are the "grands Villages" of the Panis, the Wichita people. Further northwest are the the N'ihata, called the Comanches by the Spanish, but the French used their native word for themselves. Interestingly, the Red River region is not denoted as any European-derived land. It seems to be the domain of the native people, although its borders (in yellow) indicate a Spanish claim.
West of today's Texas is "Nouveau Mexique," which the French also named the Apacheria ou Pays des Apaches - the Apache Nation. East of today's Louisiana are the villages of the Natchez, Chaktas (Choctaws, or flat head) and Chicachas (Chickasaws).
Also, notice how the western borders of the now-states of Virginia, Carolina (South), and Georgia stretched all the way to the Mississippi River in 1783... this is one of the reasons for the bicameral legislature and the electoral college in the U.S. constitution, FYI.
While this map recognizes the native claims on the lands that surrounded the Red River, U.S. maps from the early 19th century mostly depict this area as uninhabited. By the time U.S. Americans came to the region, especially along the Red River that divided Texas from the Louisiana Territory, the native people had endured decades of disease, war, encroachment, and intermarriage, which forced them to become nomadic (especially the Panis) or assume new American identities. For better or for worse, this is the result of human movement.
But wouldn't it be wonderful to see a map of the Red River before European documentation? One can only dream.
