I'll probably get some pushback on this but it is what it is.
Below is link to one of my 03s (actually my first one).
M1903 657707 - Google Photos
Since then a handfull of stocks have popped up very similar to yours.
A lot of common traits are
1) Crudely cut grasping grooves. (with some variations from stock to stock)
2) Crudely cut cutout well. (with some variations from stock to stock)
3) J stamped on the foretip of the stock (I see yours has that)
4) Some have a J stamped at the top of the buttstock.
5) Lacking a lightening cut in the buttstock. (please check yours and post)
Also, I notice yours has a buttplate similar to mine which looks like incomplete checkering. During the late 1930s, the USMC ran out of checkered buttplates (this is in the USMC QM records at NARA), complained that smooth buttplates slipped against the shooters shoulder. They purchased a "checkering" machine but couldn't get it to work. I always wondered if some of these unfinished checked buttplates were USMC before abandoning it and going to simply stippling them. I don't know if this is the case but it's the only primary source that would somewhat make sense.
Lastly, the USMC was unable to obtain more M1903 S Stocks after a 1940 Springfield Armory contract for a small run of S stocks.
You can identify these by M1903 S stocks with grasping grooves and "S0" stamped in the cutout well. Like the C stocks of the same period. In 1942 they were in negotiations with RF Sedgley for replacement S stocks (in addition to other replacement parts like barrels). There was some back and forth before the correspondence stopped (because NARA only has through 1942 and Federal Records Center at Suitland Maryland destroyed the USMC QM records from 1943-1945, I just got confirmation on that last year).
These stocks show up on USMC rebuilds with M1903 S stocks without grasping grooves with a "J" stamped in the cutout well (similar J to the one on yours). Collectors sometimes refer to them a "USMC J Stocks"
Everytime I see one of these stocks (like yours) do seem to pop up on rifles with some USMC rebuild traits. There's another CMP member I met for a cigar years ago that has a very similar rifle. I don't remember the specifics but I just remember it had a lot of usmc rebuild traits and the stock was very similar to mine. I understand the Mark I receiver makes it less of a solid argument given that the USMC rejected Mark I rifles from the Army during the interwar years.
Unfortunately, we'll never know for sure because the National Archives destroyed the paperwork that would've concluded the story with primary source documentation. These are just observations. These stocks appear to be uncommon to say the least but they do surface with common denominators. Who knows.... thanks for posting the pictures!