When was my gun manufactured?

I recently acquired my fathers old 30-06. It is a Springfield Savage Super Sporter. Under this name are the numbers 3388.
Thats about all I know. I don't see anything more about model number.
My father worked for a farmer one year. At the end of the year the farmer was not able to pay him what he owed him and gave him this gun. I think this was around 1940. My father hunted with the gun for many years. I would love any additional I can find about it. I appreciate any information you can give me. It is NOT for sale.

Unsure of the year but I know the model. If it's uncheckered it's Savage model 40, if it's checkered it's a model 45.

Does "checkered" mean checking on the outside of the frame?

No it means it's got a stock set with checkering on it.

Any ideas on how to find the date of manufacture????

Best I can do is from 1928 to 1940.