A load-handling clamp mountable on a lift truck has one or more hydraulic clamping cylinders of the type having two or more sequentially extensible stages capable of applying either load-weight-responsive variable hydraulic load-clamping force or, alternatively, load-size-responsive variable hydraulic load-clamping force, without risking the clamp's dropping of a clamped load. The length of the hydraulic clamping cylinder(s) may in some applications extend generally forwardly from a supporting lift truck, and in other applications extend generally laterally from a supporting lift truck.