Just get bigger squares of material, perhaps 8 in X 8 in (20cm X 20cm), similar to those used to line the bar track where bartenders work. Or the "no-slip" thin version that bartenders use on the slipperier parts of the bar.
You obviously need the "no-slip" on both sides of the material. And getting the bar track "tread," specifically the thinner version, should resolve your problems.