Welcome aboard from the hills of ol' Wyo.
Your Chiefs Special is a dandy -- one of S&W's icons.
S&W produced the first ".38 Chiefs Special" in 1950. When the company assigned serial numbers in 1957 it became the Model 36.
Does it say "MOD. 36" on the frame inside the yoke, or is it "MOD. 36-1"?
The Model 36 of that era (late 1975 ~ early 1976) had a tapered barrel, and the Model 36-1, which had been introduced in 1967, had a heavy barrel. 1975 was the year S&W stopped making the tapered barrel 3" Chiefs, and the heavy barrel became the standard.
If you take off the grips, there should be a serial number stamped inside the right one. It should match the serial number on the butt if they're original to the gun. The "J" will not be in line with the numbers, but there might be a larger "J" stamped elsewhere on the grip.
Enjoy,
Bob