Thank you everyone for the suggestions. A nursing friend of mine gave me about 6” of co-flex bandage wrap and said to give it a try. It’s what they use in place of adhesive tape for keeping bandages on (for instance after taking blood from a patient they will wrap this around the arm to hold the gauze in place instead of adhesive tape. It sticks to itself and is very grippy. I tried it and it works great, almost exactly what I’m looking for. I was amazed at how well it gripped. The only problem is that it isn’t clear so I can’t use it for squaring up.
There’s a quilt show in my area this weekend, hoping I can find one of the above mentioned items there.