I am no expert, but, technically, I do not think you have a model 36. First of all, no model numbers were assigned until 1957, so at the very least, it is a pre-36. However, with this model, I am pretty sure that there was a significant change from the your gun to the model 36. If I am right, the J frame grips will not fit your gun. The grip is shorter than what would become the J frame. These were actually known as "Baby Chief Specials." If I am off, please somebody let me know. I have one that is serial # 35XXX, and it is from 1953 or so. Great guns, and if in decent shape, they are worth a pretty penny. 500-600. Does it have a flat latch cylinder release?
Check with s&wchad, he has got some good info on these little guys.
Help identifying a snub-nose .38