Are you sure there isn’t a serial number on the front face of the receiver (it’s not on the side)? The early barrels were serial numbered to the gun for headspace reasons. Later guns serial numbers allowed barrels to be swapped between guns (and serial number stays with gun not barrel).

Found this on Ithaca’s website, says it better than I did


Q: Where is the serial number located on my Model 37?
A: On the Model 37®, the serial number is located on the bottom right side of the receiver or on earlier models the chamber end of the receiver between the magazine and barrel.

Also stole from this thread

https://www.shotgunworld.com/bbs/viewtopic.php?f=59&t=376421

Rules of thumb about checkering on field grades. Higher grades such as Deluxe, Skeet, Trap, and Supreme have fancy wood and hand checkering stock and fore end all years:

1. 1937-1945-Hand checkered stock-hand checkered round fore end

2. 1946-1954-Hand checkered stock-ringtail fore end

3. 1955-1965 No checkering on stock-ringtail fore end

4. 1966-1974 Negative pressed checkering

5. 1975-1985 Positive pressed checkering