For years, I have saved the small, round tins that mixed nuts or fruitcake come in. The roll fits perfectly inside and keeps the duct tape clean, dry, and it protects the roll from damage if something heavy drops on top of it.
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I use duct tape for many of my crafts and with cats, I constantly am picking off hair from the edge of the tape roll. I also find that when I put it away in my storage box it sticks to everything.