Hunting antelope on the high plains and hunting elk in the high country are very different propositions, and mule deer can be found both places, so gear may vary depending on what and where you are hunting.
September in the mountains can be 80 degrees, and can be zero degrees, and everything in between. I am leaving tomorrow to do some elk hunting, and it looks like temps will be in the 70s during the day at 9000 feet, but will probably be mid 30s at night. I have been snowed in twice in mid September,only to return and have 70 degrees two weeks later. What I guess I am saying is that weather can and does vary greatly depending, and the higher the elevation, the more it can vary.
Your Texas garb might work very well for low country antelope and deer, and might even be perfect at times at higher elevations, but wool is a great choice for lots of days in elk country. For elk, good hiking boots that are well broken in will make for a much more pleasant experience, and a good level of fitness will make the hills a bit easier.