Howdy folks, I'm new to hunting and recreational shooting this year (after a long history of my rents being too afraid of guns to let me hunt), go fig, and picked up a used rifle from a private seller. Could be smart, could be dumb.

I'm having trouble identifying the rifle outright, I know at least the action/barrel is Stamped "U.S. Smith-Corona Model 03-A3" with the serial 3680085, placing the action and barrel firmly in the 1942-1943 I think, M1903-A3. It's got a Williams-side mount for a Weaver K4-F scope and a Lyman peep-hole site.

There's a P cartouche stamp on the underside of the barrel, but pulling the stock off and looking it over (thoroughly) I don't see an identification stamp anywhere on the bed or anywhere else. But for all intents and purposes it looks like the NRA sporter stock with a few modifications.

So how do I put this all together, I know I bought an M1903, but what is it (or is it just a highly customized hodgepodge?)

Can anyone help? Thanks in advance too, if you need pics, I can try to figure out how to get em up on here.