More information and photos would help but an initial guess would be a .32 Hand Ejector Third Model (round butt) OR .32 Regulation Police (square butt). Both models are circa 1917 - 1942 with serial number range 263001 - 536684 Serial # might extrapolate to the 1920's?
Assuming I am correct: It would be built on an I-frame, smaller than a current J-frame. It would have 5-screws. Barrel length measured from front of cylinder to end of barrel would originally have been 3-1/4", 4-1/4", or 6". Cylinder number should be on the butt, cylinder, and underneath of barrel under ejector rod. There should be a small S&W trademark on the left side of the frame towards the rear. I would guess based upon serial number it would have a single line address on the right lower front frame but as best I can tell the I-frames had 4-line address before WW II (???) so it could have a 4-line address (???) ---- hopefully a more experienced member can come along to clear that up. Should have fixed sights with round front blade but could possibly be a target version with adjustable rear sight (that would be nice)?
That .32 would be from right around 1920. It would not yet have "MADE IN USA" stamped on the right side of frame. Nor would it have the four line stamp including S&W address, which was only after WW II.
Does it have the original grips (serial # on back side of right grip)?
If it has wood grips with gold plated medallions, the gun is from just before 1920. If wood grips with no medallions from after 1919.
Closest on my SN list is 318xxx, which shipped in 7/1920. Yours is definitely a .32 Third Model from before mid-1922, when the MADE IN USA frame stamping began. I'd guess 1921.