It really depends on who you know--my cousins are all cowboys but their names don't all scream country. Some do though so here are some relatives that are down-home (Canadian) cowboys.
First name that comes to my mind is Beau (or the more extravagant Beauregard).
Some other ones that have a bit of country flavor:
Levi - Love this.
Roy - I love Roy. Huge soft spot for it. Not too popular, distinctive but not crazy. Would also make a good middle.
Lucas (nn Luke) - I like this. Very popular though.
James - Goes with everything. Not especially country but not NOT-country. Very VERY boring as a middle, okay as a FN if you want.
James is inoffensive, Lucas is nice, Levi and Roy are my favourites.
Hank, especially short for Henry or Hiram
Isaac called Ike
Ezekiel/Hezekiah called Zeke
Wade, Cade, Wyatt, Grey, Blake, Hank, Clint, Clayton, Cash, Randy, Montgomery, Allen, Alvin, Jake, Garrett, Grady, Beau, Rhett,
Someone already mentioned Amos and Jesse. I'd consider Jedidiah nn: Jed, meaning "friend or beloved of God." (I personally know one teenager of this name who went by the Geek-chic "Jedi" for a long time.) Tom or Tommy was pretty popular in the backwoods of Alabama, while I lived there. Also Chase and Cole.
Thank you! These are the kind of names I want. I wanted to avoid the in-your-face country names. The kind of names that you suggested are more my style.