What caliber? Also, do you have additional pictures on the condition?
I would suggest google on "pre-64 -caliber- for sale". see why pops up. There are big spreads on these and a lot of factors, mostly condition. There are a few variations in some model years that can make them more or less valuable and there are certain barrel markings you can look for to aid in your search to distinguish between one vs another of the same caliber.
I recently went down the pre-64 rabbit hole. I learned a few things and got a general range where I think my rifle sits. I am not selling it, so was happy to just get an idea.
the "Winchester" brand stamp has variations based on years and run of a particular rifle and can help identify what exactly you have aside from the serial number also.
Check out winchestercollector.org and their forum pages. Lots of good reading for learning about the rifles.