3/8" Rod? My .58 loads just fine with it's factory 3/8" rod & I don't use a muzzle protector - no need to. I guess some of the guys might use one when cleaning, but not when shooting at the range, on the trail nor when hunting. A tapered 3/8" to 7/16" or 3/8" to 1/2" would work fine for larger bores if the pipes allowed and not need a muzzleprotector.
If one is required, then carry it in your pocket.
BB's suggestion about wedging the rod works.
The thong can be tied around the barrel in front of the sights with or without a small diameter string, to run under the rod to keep it snug, then it's not when you pull the rod out for loading.