The govt contracted Remington to make M1903's in the 30's IIRC. With Springfield making Garands, Remignton kept making M1903s after the war started. Remington figured out a way to make the rifle cheaper and faster, and that became the M1903A3. M1903A3's were made by Remington and Smith Corona.
Govt rifles were made by Springfield, Rock Island, Remington, and Smith-Corona. It says which right on the receiver. It's recommended to not fire the low number Springfields (under 800,000 for Springfield and under 286,506 for Rock Island) because of the way they did the heat treating resulted in some KB's. They changed the heat treating at those serial numbers. Later guns are considered "high number" and safe to shoot. Remington and Smith-Corona came well after any of the heat treatment problems and are safe to shoot.
I'd avoid any rifle with a receiver made by anyone other than the four USGI makes. They are all cast, and some are just OK and some suck. Either way, they are just parts guns built on questionable receivers and usually aren't worth the asking price. For all practical purposes, the civillian/aftermarket made guns are just a parts kit.