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.