Surfaces designed so that drops do not adhere to them but instead bounce off have received substantial attention because of their ability to stay dry, self-clean and resist icing. A drop striking a non-wetting surface of this type will spread out to a maximum diameter and then recoil to such an extent that it completely rebounds and leaves the solid material. The amount of time that the drop is in contact with the solid--the 'contact time'--depends on the inertia and capillarity of the drop, internal dissipation and surface-liquid interactions. And because contact time controls the extent to which mass, momentum and energy are exchanged between drop and surface, it is often advantageous to minimize it. The conventional approach has been to minimize surface-liquid interactions that can lead to contact line pinning; but even in the absence of any surface interactions, drop hydrodynamics imposes a minimum contact time that was conventionally assumed to be attained with axisymmetrically spreading and recoiling drops. Here we demonstrate that it is possible to reduce the contact time below this theoretical limit by using superhydrophobic surfaces with a morphology that redistributes the liquid mass and thereby alters the drop hydrodynamics. We show theoretically and experimentally that this approach allows us to reduce the overall contact time between a bouncing drop and a surface below what was previously thought possible.