That's a dandy find. I see why you use that screen name.
And a square butt, to boot! Congratulations.
It appears S&W might have shipped your Chiefs Special Airweight
near the beginning of spring 1957.
On June 12, 1957, S&W issued an order assigning model numbers
to its revolvers. By September 1958--the earliest known ship
date for numbered Chiefs Specials--younger siblings of yours
would become the Chiefs Special Airweight Model 37.
They were known for a cracked frame above the yoke cut, where
the barrel goes into the frame. Check for that.
Also yours comes from the era where S&W began to delete the
serial number stamps on the bottom of the barrel, back of the
cylinder, and back of the yoke (seen through a cylinder hole).
The inside of your right stock should be serial numbered.
Enjoy that little Chiefs Special. You've nosed out a prize.