I’ve done something akin to the collapsed roll.

-I started with a sheet of stickers, as used in a laser printer (I regularly use these, but you could use something else)
-Once I’d removed the stickers I was left with a sheet of “paper” which has a shiny size where the stickers were attached.
-I wrapped the paper around a round handle and cut the paper roughly to size. I used a tiny piece of tape to hold it in place.
-Then I removed a section of tape from the roll and carefully wrapped it around the sticker paper (still on the handle)
-Then I slid the paper with tape off the handle, trimmed it again and easily squashed it flat.

This way I don’t need to tie up a card with tape and the tape is usable right to the last cm.