I came across a reed today that I had replaced the paper coverings on the edges with masking tape. I knew when I did the repair that it probably wouldn't last long, since masking tape tends to get brittle pretty quickly.
The tape was not sticking, so I decided to try gluing it back down with Tacky Glue. It sort of worked, but not entirely to my satisfaction because it rippled a bit and I had to keep pushing on it to make it stay.
I started looking for something else that was thin to try instead and decided to try parchment paper, the type used for covering baking sheets. It is found in the grocery near the wax paper and foil.
I cut a small test piece, enough to wrap around from front to back, put some glue on the paper, folded it together to smear the glue around and then wrapped it around the edge of the reed. It stuck with no trouble, created a few wrinkles which I was able to remove and then I left it to dry.
I checked it a few minutes ago and it is firmly glued to the reed edge with no indication that it is going to peel off the reed.
This is definitely the way to go to replace old flaking paper covers. It is inexpensive, thin, easy to cut with a rotary cutter and ruler, will actually stick, and looks like the repair should last for a long time.
An added bonus is the remainder of the parchment paper can be used in the kitchen. I love using it because I dislike trying to get baked-on grease off my cookie sheets. I never have to do that when I use the paper.