Most interesting 300 H&H. As Fullmetaljacket stated, you guy is mid 1937. I have attached a most interesting historical article about how the Wimbledon Cup was won in 1935 by at 300 H&H, first time for a magnum case to win the 1000 yard match. This would have been the seed to have your gun built in 1937 at the Winchester factory. The date will be stamped on the bottom of the barrel just in front of the receiver being "37" and most likely also stamped in the stock near the same place.
I believe you will have difficulty in obtaining a valid appraisal on this one. It is not completely stock, which hurts the collector value, but it is a rare bird. The military sniper rifles ordered by Van Orden and saw duty in many places in the 1950s have created a huge demand, up price tick in pre 64 target rifles.
The guys in Woodville, Washington are the experts in these pre 64s and are most likely your best bet for obtaining maximum funds for a sale. Look at their blogs and you can educated yourself some about target models, the Van Orden guns, etc.
Someone said the 300 H&H is obsolete, I do not agree. Fantastic round, yes I have a late 50s, pre 64 hunting gun in 300 H&H. Winchester chambered over 37,000 of these pre 64s between 1936 and 1963. It is not a rare chamber. And then some dude by the name of Roy Weatherby out in California in the 1950 decided to follow the idea of P O Ackley and improve the case by blowing out the sides and called it the 300 Weatherby Mag. Nope, they will never quite making 300 H&H brass.
some of the photos I saw a few minutes ago show that block siting on the back of the receiver bridge with a peep sight mounted, so it is possible this was on the gun when it departed the factory in 1937. This gun would have been at the top of the heap if one wanted a long range competition gun in 1937, as very few custom shops were out there.
Check the guys at pre 64 win dot com and look at the blogs and you may want to send photos for help. Attached is the article about the world waking up to one of the best 1000 yard cartridges to compete and win at Camp Perry over the years.
Wimbledon Cup History: Ben Comfort's Controversial Sighters | An NRA Shooting Sports Journal