Originally for the 1903, it held over for the M1 when they started. The M1 used all early M1903 stuff originally. The container end may have a thin thong pull through and a nylon bore brush.
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Your Remington M1903 was probably overhauled and re-barreled by the Navy. That's why it exhibits no overhaul inspection stamp.
The pinned stock on your Remington M1903 is a replacement from a much later Remington M1903 or an early M'03-A3. The very late Remington M1903's were fitted with pinned stocks.
The stock on your M'03-A3 is the correct and probably original stock. Late Remington M'03-A3 stocks were fitted with crossbolts. I would definitely NOT recommend exchanging stocks.
Thanks for sharing! And Merry Christmas!
J.B.
Congratulations on a nice "barn" find.