The invisibility cloak, a long-standing fantastic dream for humans, has become more tangible with the development of metamaterials. Recently, metasurface-based invisibility cloaks have been proposed and realized with significantly reduced thickness and complexity of the cloaking shell. However, the previous scheme is based on reflection-type metasurfaces and is thus limited to reflection geometry. In this work, by integrating the wavefront tailoring functionality of transparent metasurfaces and the wave tunneling functionality of zero-index materials, we have realized a unique type of hybrid invisibility cloak that functions in transmission geometry. The principle is general and applicable to arbitrary shapes. For experimental demonstration, we constructed a rhombic double-layer cloaking shell composed of a highly transparent metasurface and a double-zero medium consisting of dielectric photonic crystals with Dirac cone dispersions. The cloaking effect is verified by both full-wave simulations and microwave experimental results. The principle also reveals exciting possibilities for realizing skin-thick ultrathin cloaking shells in transmission geometry, which can eliminate the need for spatially varying extreme parameters. Our work paves a path for novel optical and electromagnetic devices based on the integration of metasurfaces and metamaterials.