Snapshot hyperspectral imaging can capture the 3D hyperspectral image (HSI) with a single 2D measurement and has attracted increasing attention recently. Recovering the underlying HSI from the compressive measurement is an ill-posed problem and exploiting the image prior is essential for solving this ill-posed problem. However, existing reconstruction methods always start from modeling image prior with the 1D vector or 2D matrix and cannot fully exploit the structurally spectral-spatial nature in 3D HSI, thus leading to a poor fidelity. In this paper, we propose an effective high-order tensor optimization based method to boost the reconstruction fidelity for snapshot hyperspectral imaging. We first build high-order tensors by exploiting the spatial-spectral correlation in HSI. Then, we propose a weight high-order singular value regularization (WHOSVR) based low-rank tensor recovery model to characterize the structure prior of HSI. By integrating the structure prior in WHOSVR with the system imaging process, we develop an optimization framework for HSI reconstruction, which is finally solved via the alternating minimization algorithm. Extensive experiments implemented on two representative systems demonstrate that our method outperforms state-of-the-art methods.