The M1903A1 was adopted in 1929 and it was basically a "regular" M1903 with a Type C (or later Scant) stock, whether "new built" or overhauled. Some service M1903A1s were built in 1935 and afterwards. The Type C stock was also used on nearly all National Match rifles after about 1928.
Those selling rifles will often describe almost any later M1903 as a "1903A1" when they weren't. The Remington M1903, for example, wasn't a A1, although some sellers describe it as such. The Remington was produced as manufactured with a straight stock, not a Type C.