Hello, I usually get to the party late but I think I'll give this a try. Source-"History of Smith & Wesson" by Roy G. Jinks.
When you open the cylinder is there a number in the yoke area-under the barrel. This would be the model number, if its a round butt it would be a 30, a square butt a 31. These numbers stared in 1957.
Within the 30 model series--There was a "change ... authorized on December 13, 1955, beginning at approximately serial number 640,980". Serial numbers ran 826,977 completed in 1970, with a new number range beginning with an H prefix. So as a guess with existing info '55-'57. More information is required for a better approximation. I could of course be completely wrong.
So I may have answered your question and I may not have. Open the cylinder and look for the model number and does it have fixed or adjustable sights, round or square butt.
There will of course be a price to pay for this info, you must include at least 1 clear photo, as they say "without a picture it never happened". I'll check back if no one else has a better answer