The stock is indeed a late Remington M1903 stock (not M'03-A3). The flaming bomb on the fore end tip appears on all Remington stocks from that period. I don't recall having seen the "S" marking before, so one can only speculate what it means. But it is not a standard Remington marking.
Nice rifle! Thanks for sharing!
J.B.
p.s.,
Your rifle also appears to be fitted with a Smith-Corona M'03-A3 bolt and an M'03-A3 trigger and stacking swivel.