Chris is correct. I have been loading my jaeger with its 3/8" parallel hickory rod since I made the rifle in '97, and without a problem. And you must know that we up here in the Great White North use tight combinations, and a starter. This rifle is .60 cal with round grooved rifling. The mold I have is a Lyman .595" but it throws balls that measure .598" And I use a .020 - .022" denim patch. Once that combination is in the bore, the 3/8" wooden rod is all that's needed to put it firmly on the 86 grains of FFg.
You don't want a 7/16" rod hole in you rifle...it'll make the forend necessarily too fat or deep vertically. Stay with the 3/8" hole and rod.
For target shooting, I have made a 3/8" steel rod just to give the muzzle some weight, and it makes loading a breeze for sure, but it is not needed.