As a general rule, the cowboy of the era didn't carry a canteen, as is so often seen on the Silver Screen.
There was a water barrel on the wagon if he needed a drink, and there were streams and creeks that criss-crossed the region that have long since dried up that were his mainstay.
They didn't feel any pressing need to 'stay hydrated' as today's folks do.
'If' there was a canteen - the likelihood of it being ex-military would be high, since they were the cheapest thing to buy.
Cowboys often carried a few 'airtights' in their saddlebags - chiefly canned tomatoes - to provide both drink and sustenance (that is, if he wasn't afraid of tomatoes, since many folks called them 'love apples' and thought them to be poisonous).
If you need more water - and have a way to transport it - you could use a stone jug.
Or - you could seek out a 'canteen gourd' and make your own canteen, as was done in the Southwest by the Vaqueros.
(A 'canteen gourd' is an actual type of gourd - large and sort of 'canteen-shaped', and well-suited to the task.)
They'd hollow out said gourd and let it dry, while they'd fashion a harness and strap to attach to the pommel, then fit it with a cork or whittled wooden plug, and they were in business.
Were I to do that, I'd make certain that the insides were scraped as clean as possible, re-scrape to make sure, then I'd coat the interior with beeswax.
Good Luck!
Vaya,
Scouts Out!