A highly efficient thermal rectification applicable to large panels still needs to be developed. Here, we experimentally achieve a high thermal rectification efficiency of 33% by carefully engineering elastic modulus asymmetry in a centimeter-scale bilayered silver-graphene oxide sponge. The thermal conduction primarily occurs in the out-of-plane direction, and the forward heat flow direction is from the hard silver to the soft graphene oxide. Surprisingly, the forward heat flow direction is reversed when a silver layer is formed on a harder polystyrene foam. The forward direction is always from the harder side to the softer side, and the asymmetry in elastic modulus is suggested as a possible mechanism based on the one-dimensional Frenkel-Kontorova (FK) model. The finite element analysis indicates that other mechanisms such as temperature-dependent thermal conductivity and radiation asymmetry cannot explain the high rectification efficiency. This scalable work over a wide temperature range may find immediate industrial applications.