Hi, and welcome to the forum! I've gotta warn you though, this place . . . and S&W revolvers in general is/are addictive!!!
Yep, I suspected yours wasn't a mid/late '50s version because it lacked both the square cylinder release latch and the "diamond" stocks of the that era.
You've got a "floating J" (serial number) M37. So is my "always" (on me) CCW snubbie, though mine has the round butt.
They are great little guns and very fun to shoot. I recommend getting a Tyler T-grip (see my photo) to use with your stock grips and you'll be quite pleased with how nice and comfortable it is to shoot.
I practice with 148 grain wadcutters usually but if your revolver is in top shape an occasional cylinder of +P shouldn't hurt it at all.
Congratulations on your '71.
My original stocks were as new, so I removed them and put 'em in the safe . . . and replaced them with forum member Patrick Grashorn's gorgeous Elk stags. I've got Grashorn Elk stags on several of my favorite Smiths that I shoot a lot. Durable and beautiful too!
BTW, before '68 the stocks (S&W's name for their grips) had a diamond in their middle, such as this Model 36 from 1964 below. Also, after about 1966 they replaced the "square" cylinder latch I mentioned earlier with the kind that is one our Model 37s.
The Model 37 is the aluminum framed brother of the steel framed Model 36. Here's another shot of my M36 before I put a Tyler T grip on it too . . .
Love those vintage Smith snubbies. Enjoy yours!!!
PS: My wife's CCW revolver is also a Model 37 from 1971! Her's has a square butt like yours but also has a 3" barrel. She shoots it really well too!