Own a low number 1903 Springfield made about 1915, but barrel is 1920. Acquired it about 4 years ago and it'd been well used by previous owner(s). As its barrel, sights and action was untouched and only its stock butchered, (did have a broken extractor collar), restored the rifle to its military condition with a Boyd 1903 stock and G.I. metal parts for the stock. Usually shoot the rifle in our private ranges monthly surplus rifle competition. Could shoot a darn nice condition 1903A03 Smith Corona in the surplus competition, but prefer the rear sight of the 1903.
As to what available factory ammunition would afford the lowest chamber pressures? Think Remington may sell reduced recoil 30-06 ammo.
As for me, LOL, standard 30-06 loads in any of my Springfield military rifles are not pleasant for me to shoot from benchrest. But for safety reasons due to my 1903 Springfield being a low number, shoot only my reduced reloads in it. Prefer shooting my IMR Trail Boss powder reloads. Use about 19 grs of it with 125 or 135 gr jacketed bullets. Velocity is between 1400 - 1500 fps with them and accurate enough for me to use in the surplus rifle 100 yd competitions at my range. Loaned a few of my trail boss loads to a fellow competitor to try in his 1903A03 after one of our competitions. Guy hunts in Africa and definitely has some high recoiling rifles and he admits regular 30-06 ammo kicks him pretty hard in his 1903A03. My 30-06 trail boss loads, hit about 8" lower on his 100 yd target than where his 30-06 reloads would have hit, but grouped pretty good for him.