In short - no - there was no provision for an enlisted soldier to carry a private-purchase weapon at the time.
Officers did - but they could purchase theirs from the Army as well as from Smith & Wesson, and they would've been the long-barrelled model, just like that in general service.
The short-barrelled Schofields came about after the guns were declared surplus, and sold off to outfits like Schuyler, Hartley and Graham, were refurbished, had their barrels shortened, were generally nickel-plated (though not always) and were sold to the public, or to firms like Wells Fargo and various Railroads for use by their guards.
Vaya,
Scouts Out!