I've used velcro and various thicknesses of leather. Hard to get less than 1/16" thickness with those, you can get thicker with leathers... used soft splits, brain tanned, and smooth.
Recently though, I've taken a scrap piece of shearling, trimmed down the pile side with sharp scissors, and it's become my preferred material. A little on the thick side, but soft and quiet.
Also used/use a toothpick, matchstick, etc. on some set-ups. I'm preferring to just go with the radius cut on the bow lately.Like the double sided tape.