Welcome to the forum.
Congrats on acquiring such a wonderful family heirloom and in such good shape as well. That correct box adds about $100 to its value. Your Grandmother may still have its original cleaning tools and paperwork that came with it in Granpa's things, that adds even more value.
DWalt gave you the official model name above. Collectors describe it in more detail for clarity of its variety as the 32 Hand Ejector Improved I frame – 1st version (with ½ penny front sight) with round butt. I agree, 2/53 is the most likely period it shipped from the factory.
The post war and 1950s was a very evolutionary period for S&W revolvers and yours is no exception.
32 HISTORY
The .32 Hand Ejector and its cartridge, the 32 S&W Long are both historical. The .32 HE is the first side swing out cylinder produced by S&W, built on the first I size frame, and introduced as the 1896 Model (1st Model) along with its new cartridge. Production of this model extended to serial # 19,712.
It went thru several evolutionary changes until it became the 32 HE 3rd model in 1917 and also when a sq butt version was introduced as the 32 Regulation Police beginning at ~ #258000 in the same serial range as the 32 HE, with a most common barrel length of 4".
Target models with adjustable sights were available with 6" barrels as well as very rare 4 or 4 1/4" barrels, and with 2 screw extended target stocks.
The 3rd Model had a hammer safety block added around 1919 used thru WW II.
By 1920 the cyls received a heat treatment for additional strength as did all S&W revolvers.
Both were reintroduced after WW II in identical form except for the change to the much more fool proof post war sliding bar safety. They went thru more evolution in the 1950s, and in 1957 became the Model 30 w/round butt and the Model 31 w/square butt, and finally in 1961 were upgraded to the slightly longer J size frame with added -1 to the model #s.
Eventually both versions were combined as just the Model 31-1 in 1976, until discontinued in 1991. Various iterations were later introduced using the 32 H&R Mag and/or the 327 Fed Mag, both of which still use the 32 Long as well.
Any currently available 32 Long ammo is perfectly safe to shoot in your 32.
