2152hq, thanks for the continuing education. I knew that was the Vienna mark, and I recognized the double eagle. I just didn't know that the one began to be applied so late in this model's production.
I saw a photo of a post-Anschluss M-S whose top stamping said "Made in Germany" rather than "Made in Austria"! That may have been produced later in the war, but I suppose it could have happened as early as late 1938 or in the course of 1939. Mine says "Made in Austria," but I think the English language stamping started in the early '30s or even '20s. I haven't seen enough to know.
I should have said that on the underside of the barrel are proof marks +1 and 1, as well as 3B with a punch mark following, and C6.7; I take it this last is an alternative description for the 6.5 x 54 round that emphasizes the groove-to-groove dimension of the bore, rather than land-to-land. Enlarged in a photo, what I thought was 3B-dot looks more like 36-dot.
There was also a small ornate stamp that looks like a script E over V, with branches on the top of the V like antlers.
[EDITED TO ADD; I just found out this is another "Vienna" proof designation.] I tried to get a photo:
In addition, the stamp "33." appears under the receiver about an inch away.
I have resisted thinking that the 33 or 36 could be a year stamp for a couple of reasons, the obvious inconsistency being one. There is a 1934 M-S with a serial number just over 15000. I don't know whether the Austrians ever numbered production out of order, or made more than a couple of thousand rifles a year, but it just seemed to me that this gun, at 20953, had to be either very late '30s production or, assuming postwar resumption of production on this model even happened, late 1940s or very early 1950s. I have been unable to determine when the M1903 went out of production for good. I don't know what the highest serial number on a 1903 is.