I had the same thing happen to one of mine. Actually it was a metal plate that had the double-sided tape which the hygro then stuck to with a magnet. Anyway, I just glued it back to the same spot with Gorrilla Glue which doesn't seem to have much of an odor.
You just wouldn't want to use any type solvent to remove the tape residue. Unless you were going to do a review which might read: "This cigar had a spicey finish with just a hint of Naptha and Xylene."