Here's a teal Southwestern chair if it gives you inspiration: https://www.hometalk.com/8835191/a-southwestern-chair-makeover?expand_all_questions=1.
I would think cream or gold would work well.
Jeanie, here are a few ideas that might spark your creativity and likes. Hope this helps. The first one is a pallet of colors that go with it.
Here is a colorful rug to give you some inspiration. I love this rug.
Terra Cotta works very well. You can pull in that color with flower pots.
I think teal and a burnt orange go well for a southwestern look.