*Thinking of possible useful plastic items*
-coffee (or other large) can lid, maybe cut 2 and double up
-lid from microwavable container that is no longer used
-plastic storage container that shouldn't have been microwaved that last time
-the dreaded plastic shell packaging if you could manage a piece it of unmangled and large enough to use
If the reason for the tape is to make it slide in easily, I think you might just use something smooth to temporarily wrap it and then remove the wrapping. If it's for the edges that are sharp/rough, I wonder if there might be something you could use to melt them a little and smooth them out. I have an old woodburning tool that I've used for melting holes in plastics for potting plants, but I don't know how that would work for smoothing edges.
Happiness is catching that dropped stitch while it's still fixable..