The iconic Winchester rifle didn’t just pop up out of nowhere. It’s a descendant of the 1854 Smith & Wesson lever-action rifle, which is quite similar to the Henry Rifle (another icon in its own right).
Oliver Winchester invested in Smith & Wesson in 1855, shortly before Horace Smith and Daniel Wesson reorganized the company and ceased production of the rifle. Winchester came out with its first rifle in 1866, right after the Civil War.
These S&W rifles wouldn’t be common, but your Far Western character could certainly have one. (It probably wouldn’t have been as fancy as this one, though; this was probably a showpiece meant to attract buyers and investors.)