Mesquite is good if you can find a stave worth working.
I live in Utah, so we share woods, but not all or entirely. You may have to go UP, as in higher elevation to find more scrub woods like serviceberry.
Look for "feral" trees at the edge of civilization, too. Defunct farms often have plums, cherries, apple, or other fruit and ornamental trees. Elms spread fast and seem to be everywhere; vacant lots, watercourses, alleyways, riverbottoms, that field that is about to be developed into houses, etc. My home county road dept is always cutting down elm and mulberry growing too close to the road, in ditches, and all up in the chainlink fences. When they do, it's basically coppicing, and the suckers shoot up straight and tall and can make a bow in just a few years. Black locust is often planted as a reclamation tree because it builds and adds nitrogen to soil. Ash behaves like elm, spreading everywhere, just not as much or as fast, and generally grows more slowly. Scrub maples in the mountains, like canyon maple, is a very hard, tough maple, almost acts like sugar maple.
Keep an eye out for tree workers cutting species you like, and people doing landscaping. I have found staves at the dump where they grind up green waste for topsoil. Ask around and spread the word, and see what comes back to you.
