Large thermal changes driven by a magnetic field have been proposed for environmentally friendly energy-efficient refrigeration, but only a few materials that suffer hysteresis show these giant magnetocaloric effects. Here we create giant and reversible extrinsic magnetocaloric effects in epitaxial films of the ferromagnetic manganite La0.7Ca0.3MnO3 using strain-mediated feedback from BaTiO3 substrates near a first-order structural phase transition. Our findings should inspire the discovery of giant magnetocaloric effects in a wide range of magnetic materials, and the parallel development of nanostructured bulk samples for practical applications.