Nice rifle. Several comments come to mind.
I’d clean the bore down to bare metal with Sweet’s 7.62 solvent or something similar. There’s no telling how much fouling has accumulated in there over the years and solvents now are a lot better than they once were.
30-06 ballistics have improved over the years. It earned its reputation before chronographs were common so loads were more like the 308 than what the factories claimed: 2,850-ish with a 150 and a bit north of 2,600 with a 180. Your rifle will probably shoot well with cheap factory ammo, much of which is loaded to that level. With careful handloading, the 30-06 will run almost with a 300 H&H, but I wouldn’t try that in a 1903. 55 grains of IMR 4350 and a 180-grain bullet usually runs 2,650 fps in a 24” barrel and provides all the accuracy a given 30-06 can muster. You don’t need premium bullets at that velocity and cases will last until you get tired of looking at them.
A fair amount of foreign FMJ ammo that the CMP released onto the market is still available. I’ve shot a few thousand rounds of the Greek stuff over the years, and it’s fine. Avoid the Korean. Any commercial FMJ ammo loaded for the Garand will be a hoot in a 1903. Before the pandemic weirdness, you could buy cases of it at a discount.
Try that scope before you toss it. Bushnell optics of that period were actually decent though yours may have seen some wear and tear. Older Weaver and Redfield scopes are solid, inexpensive, appropriate for that rifle, and have enough tube length to work well on it. A fixed 4x would be perfect in my book.
The 1913 Picatinny rail system is based on the Weaver rings and bases that are on your rifle. They’re strong, cheap, and they return to zero. Mounting a scope in them is a drag because the tube tends to rotate as you tighten the screws. Just start with the scope tilted a little in the opposite direction and it will line up as you crank it down. After 50 or so years, Weaver finally addressed that with these: https://www.weaveroptics.com/rings/d...s/1000239.html
Recoil pads area great place to screw up and it doesn’t take much to ruin the lines of a nice rifle. I’ve seen so many done so badly that I’ve always paid someone who understands the problem to get them right. Typical mistakes include squaring off the toe, not rounding the edges properly, and not aligning the heel with the comb, all of which you can see in picture 13 at https://www.gunbroker.com/item/885715204. You can also screw it up by leaving little flat spots so it looks like it was done by some guy with a pocket knife and a hangover, as seen in picture 21 here: https://www.gunbroker.com/item/880960132 Here’s a gorgeous rifle with a poorly executed pad: http://www.hallowellco.com/kimber_of...ica%203006.htm If it just had another 20 minutes of work by someone who understands how recoil pads should look, then it would look like this one, which I think is perfect: http://www.hallowellco.com/dakota_mo...16%20R0189.htm
Enjoy your new/old rifle.
Okie John