The Southwest is interesting because it represents two major textile traditions colliding.

First, the various Pueblos had a tradition of weaving cotton, which they had picked up from trade contacts far to the south (see /u/Mictlantecuhtli). Prior to Spanish contact, men and women in the Pueblos would have been predominantly been wearing breechclout and mantas, respectively, made from either cotton or tanned leather (pronghorn or deer). Of course styles varied, and depending on where you where ponchos and skirts might be more common. Cotton played an important part in the culture of the Southwest. It was traded west into coastal California and east to at least the lower Mississippi Valley. The cotton trade may have reached all the way to the Atlantic coast; during the de Soto entrada the Lady of Cofitachequi was reportedly carried in a palanquin draped with cotton. However, there's not yet archaeological evidence for cotton that far east at the time, so the chroniclers may have misidentified the textile. In addition to be important economically, cotton had a ceremonial context, as its fluffy white seeds evoked the clouds and, by extension, the katsina.

Second, there was, in fact, a pre-Spanish tradition of wool textiles in the area. It was likely brought down from the north by the Navajo and the Apache during their emigration from Canada. Before the Spanish brought sheep, the production of wool was a time-consuming process that couldn't produce large quantities of textiles. The main sources of wool were wild mountain goats and dogs. Mountain goat wool was collected from the clumps shed by wild goats. As for dogs, in coastal Canada there had been a specialized breed of wool dog, which is now extinct. While the Navajo and Apache didn't bring such specialized dogs south with them, the hair of less specialized dogs were still used. The introduction of sheep made wool much more readily available and easier to produce. Sheep were also easier to manage than wool dogs, which brought about the end of that bred once sheep wool products became common that far north.

In addition to leather and hide clothing, other textiles you would have been able to find in pre-Columbian North American would have been made from yucca, dogbane (aka silk grass and Indian hemp), cedar and mulberry bark (the mulberry bark cloth might have been what the Lady of Cofitachequi was actually using), rabbit fur, moose hair, and bison hair.

For Mexico, at least, clothing was made from spun cotton or maguey fibers. There were still people who made clothing made out of hides and in the Florentine Codex the person Sahagun was interviewing for this information called them barbaric for doing so. Usually the lower class people wore clothing made from maguey fibers while high class people wore cotton. The cotton clothing for nobles would then be supplemented by feather headdresses or perhaps a jaguar pelt depending on the people.