I've had a collection of dying, dead, and really dead cutting mats in my drawers ever since I bought my first electric cutting machine, a Silhouette SD. They have been waiting there all these years for a resurrection because I loathe trashing plastic and knew I'd find a method of re-applying an adhesive that I'd like some day, and that day is here.
I have tried using 2-way Zig glue, but found residue on my paper, even after it had dried for days. I tried using large sheets of double-sided adhesive (courtesy of a local signage store who was throwing it out), but I would inevitably cut through that layer and would end up having to peel it off my own cut pieces. A friend showed me a video of someone diluting Aleene's Tack It Over and Over Again with water and brushing it onto her mat.
I decided it give it a try, but without dilution nor a brush. I wanted to smear the glue, just like I always do when I'm quilling. I was so pleased to see it work so well! I was able to re-use my mats and no adhesive came away with my paper. The adhesive is smooth and grips card stock almost too well.