Lynn,
When I ream, I use various sizes of circular files that have had their points made sharper and the butt end of the files chopped square and the butt ends ground so that when chucked into a drill run true without wobble.
I start with the smaller file and then with the drill in reverse to avoid having the file pull itself into a grip or ring and splitting it- I force the spinning file into the grip to enlarge and ream the ring or grip.
For myself, I have found that the clue to insure that the reel is being reamed concentrically is to wear a knit glove on the hand holding the ring, grip or reel seat. Then, when reaming, let the gloved hand slip a bit so that the object being reamed is slowly rotating in the slipping gloved hand. By keeping the object rotating, the hole self centers and the hole remains concentric.
Best wishes.
p.s.
I use the same principal when center boring a grip or other object on my lathe. i.e. I chuck the piece being worked into the head stock chuck. Then, the drill bit is chucked into the tail stock. The lathe is turned on, the work piece spins, and then the stationary drill bit is advanced into the spinning work piece by using the tail stock advance dial or lever. Since the drill bit is stationary and the work is rotating around the drill bit, the hole has to be centered in the work piece.
For example, this picture illustrates a rear grip being center drilled for the hole for the rod blank:
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