You can pull off the old taped fletch and refletch.
Not certain if I can articulate this... for those having trouble with removing the backing.
Stick the tape to anything. Pick up the backing so the tape separates. Now, cut the tape, but not the backing. Leave a 1" (or more)tag of the backing with no tape. You can see thru the backing to where the tape ends so placement on the next feather is not a problem.
Put feather in clamp, apply tape to feather and, using 4 hands, ( I lay the clamp between my knees) pick up the tag end and pull it away from the tape until just past the end of the feather. Cut the tape.. but not the backing, and then cut the backing, again, leaving 1". After you do it two or three times, you get a feel for it and it is easy.
ChuckC