If your S&W does not have a lug under barrel to hold the front of the ejector pin, then you have a Model 1899 Army-Navy, or now known as 38 Hand Ejector Military & Police First Model.

These revolvers were made from 1899 to 1902, with about 20,000 total manufactured. Original barrel lengths included 5" and 6", so maybe yours is closer to one of those than 5 1/2". Measure from the face of the cylinder to the muzzle to check.

We would be interested in seeing a picture of the gun, since it would help with ID. Also, do your grips have a metal S&W medallions? If not, they were probably an after-market addition, which was quite popular in the early 20th century.