I also strongly recommend NOT using a sectioned rod for cleaning, especially the poorly fitting GI rod.
BUT, you can safely use the GI rod IF you PULL it through the bore instead of pushing it.
In the field, I carefully put the rod down the bore, then screw on the brush or insert a patch in the loop.
I soak the brush or patch, then carefully PULL the rod through the bore, guiding it with my fingers.
Since you aren't pushing the rod down the bore, it can't flex and wear or damage the muzzle.
The best tool you can have for the M1 is the late type M3-A1 Combination tool.
The earlier tool with the slotted patch holder was discontinued for two reasons.
First, the brush did a better job of cleaning fouling from the chamber.
Second, they found out that the slotted tool would score or damage the chamber.
The M3-A1 is also the easiest way to disassemble the bolt while it's in the rifle.
With the M3-A1 combination tool you can:
Disassemble and reassemble the bolt IN the rifle.
Disassemble and reassemble the bolt OUT of the rifle.
Disassemble the gas system.
Clean the chamber.
Tension the rear sight.
Disassemble the rear sight.
Use as a general screwdriver and pin punch for disassembly.
Extract a stuck cartridge.
One warning: NEVER wrap a patch around the M3-A1 chamber brush to clean the chamber. This "springs" the brass bristles and ruins the brush.
Haunt the gun shows, and check the surplus parts sellers for new M3-A1 replacement brushes, and buy a few spares.
The original GI grease was yellow-white Lubriplate.
Later this was changed to brownish Plasti-Lube.
You can still buy both in larger sizes from Brownell's and Champion's Choice.
In truth any high quality, water proof, heat resistant grease will work as well.
One of the best accessories you can buy for your M1 is a gas cylinder wrench.
This is a special aluminum wrench that fits around the gas cylinder,and is used to hold it steady while tightening down the gas cylinder lock screw.
This prevents damaging the gas cylinder or barrel splines, and keeps the system tight on the barrel.
The wrench can be bought from a number of sources and isn't expensive.
Tightening the gas cylinder lock screw without a wrench puts a tremendous strain on the cylinder and barrel splines, and this will loosen the fit quickly.