Umm... tape is applied to windows not to prevent them from breaking, but to prevent the window from breaking into a million tiny pieces and sending glass shrapnel all over. The tape sort of controls the glass shards. Of course, if you cover the entire window with tape, this could add to the tensile strength, but then it would be dark inside.

I roll over dead gift cards. But the fact is that many/most brands aren't as sticky when they come off a re-roll as they are off the original roll. Gorilla Tape is still sticky enough to be useful but some brands are useless.

I flat roll Gorilla Tape on itself, starting with a 4 inch length prior to the first fold. I make my flat rolls between 10 to 20 feet . The 10' flat roll ends at ~4.5 inches and is stowed in a 3" x5" zip seal bag. The outer bag helps keep it from getting raggy or dust stained in your kit Perfect for sm. to med.size emergency and first aid kits.
I also used to take 4 , 11" strips of G Tape and stick them to empty sheets of the plastic like backer paper onto which they put sticky labels. Fold it like a accordion. You can further fold it down in half or thirds, if your careful to insure the tape is well stuck to the backer, at the folds.

The question should be: What cause the tape to loose its stickiness after re-rolling? IMO, there are 3 factors:

1. Greasy/dirty fingers. Use plastic painting gloves (rubber, surgical type gloves may have protecting powder residue on them);

2. Dusty air in the room (or other re-rolling location). Tough one. I would personally do that in my small, 85% ceramics tiled bathroom after running hot shower for a while to settle any dust in the air volume with vapor. Another option would be to make a "clean room" out of a brand new large enough transparent plastic bag.

3. Mechanical deformations of the tape and sticky surface. Just avoid any tension while re-wrapping, and I think, a round pencil is better than a sharp-edged plastic card as a wrapping core.