If a .38 Special will go in half way, the revolver is chambered for .38 S&W, which is shorter and fatter than .38 Special.
The serial number may have originally included a prefix letter, but it is shown only on the left grip when the grip piece is revoved. With the full serial number we can date that revolver.
.38 Short Colt and .38 Long Colt are the same diameter as the .38 Special but shorter; neither would be right for that gun. .32 anything would be much too small.