For $500 you can buy a savage or ruger American with a scope that will shoot just fine for deer hunting. Keep the old gun WITH the receiver sight on and shoot it. You will be surprised at how accurate they are. I have a few that I shoot out to 300m plus at Sac Valley Range and they are every bit as capable as my scoped hunting rifles. Keep what was passed down as is and buy a modern rifle for the scope option. Anybody can shoot a scoped rifle, but it is a skilled marksman who can shoot open sights - Be that skilled marksman!
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Also your modern rifle will be pillar bedded with a synthetic stock, and have a brand NEW barrel. That 1903 may need to be properly bedded so the stock wont crack. How's the barrel and crown. After all the expense of the scope mod and bedding, the barrel might not be up to your accuracy expectations. Remember it is not a target rifle, maybe 2moa with factory loads.
That's what I've decided to do. It too nice a rifle, with family history and I want to retain that. I won't be able to shoot it though until February, as I'm going to be traveling, but I'm looking forward to seeing what it can do. Thanks for sharing!