The stock is not a C stock - it is called a "Scant". In WWII, the U.S. had large quantities of "straight" stock blanks. While the Type C (full pistol grip - see picture) was standard, the stock blanks were not "deep" enough to allow for a full pistol grip. To the "semi"-pistol grip stock was developed to make use of these stocks.
The original stock would have been a "straight" stock (see other picture). Remington was allowed to use straight stocks in M1903/M1903A3 production (One reason for the M1903s they produced being called "Modified".)
Type C stock:
"Straight" stock:
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