The Army contracted for M1903's in 1940 because they were afraid the production of M1 rifles would not be sufficient. But when M1 production proved enough to meet requirements, the late production M1903A3 rifles were never issued, going from the factory directly to storage. When many of those rifles were sold through DCM in later years, some people assumed that since they looked brand new they had to have been rebuilt. In fact they looked brand new because they were brand new.
In fact, not many M1903/A3 rifles were rebuilt. In the post-war period they were obsolete and the Army simply sold them off rather than waste the time and money in rebuilding them. So if you see an M1903A3 in great shape, don't make the assumption that it was rebuilt; it may never have been issued or fired outside the factory or depot.