The "right" bullet diameter really depends on the size of the bore if it's worn or not. My 6.5x55mm Swede for example is supposed to be .264" but I slugged the bore for cast bullet purposes and found out it's so badly worn it's .2665". It would probably like the .267" Carcano bullets even though they are the "wrong" size.
Many old, worn, surplus 30-06's prefer .310" bullets meant for a Mosin Nagant. I've had two different .303B rifles with groove diameters of .3155" and .3160". The "correct" military bullets at .312" (that's what the WW2 surplus I pulled apart all measured with a mic) are mediocre at best in both rifles. Oversized cast bullets shoot much better with smaller groups. If I could find common jacketed .316" bullets I would love to try those.
I've always found that using a bullet that properly fits your barrel will give better results than the technically correct bullet diameter in old surplus rifles. If you have a mint Carcano with a perfect bore then you will most likely want the technically correct bullet size. I'm just throwing this out there as I have no idea what condition the barrel of your rifle is in.