Most composites exhibit a damping figure of merit, a crucial index of a material's dynamic behavior, lower than the value predicted by the Hashin–Shtrikman bound. This work found that the biomimetic hierarchical staggered composites inspired by bone structure can have a damping figure of merit tens of times higher than the Hashin–Shtrikman composite. The optimum state is achieved when the hard and soft phases contribute equally to the overall stiffness of the composite in the direction parallel to the platelets. At this optimal point, the model predicts that the overall stiffness is half the Voigt bound while the damping loss factor is half that of the soft phase. This behavior stems from a deformation mechanism transition from soft-phase-dominant to hard-phase-dominant as the platelet's aspect ratio increases. The findings from this study may have important implications in the future design of composites to mitigate vibration and absorb shock in load-bearing structures.