Be very careful, but you can grip on less than .060”. You must have a good edge with minimal edge break on your material. You may lose some gripping power as the grip height decreases, depending on material. You must modify the jaws by machining the grip groove deeper to allow the grip to set lower.
CAUTION
If you machine the groove deeper you will not be able to use the stops. You will break off the stop tabs when you tighten the screws. Remove the grips, stops and screws , of course, before machining the grooves deeper. Keep the groove width the same as the existing groove, or you may damage the grip, screw or jaw when clamping with the vise.