Yes I use hide glue, I find it holds up better over time.Here is what you ask for in an auto parts store.
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Almost any glue will do with one caveat.
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I have clamped another vise cantilevered in a high vise and left it overnight and found wrinkled jaw liner the next day from the torque of the clamped vise’s overhanging weight. Contact cement or high-gram hot hide glue have resisted wrinkling the best for me. You need a hide glue that is solid at room temperature. For example, Old Brown Glue did not work.