Snapshot Compressive Imaging is an emerging technology that is based on compressive sensing theory to achieve high-efficiency hyperspectral data acquisition. The core problem of this technology is how to reconstruct 3D hyperspectral data from the 2D snapshot measurement in a fast and high-quality manner. In this paper, we propose a novel deep network, which consists of the symmetric residual module and the non-local spatial-spectral attention module, to learn the reconstruction mapping in a data-driven way. The symmetric residual module uses symmetric residual connections to improve the potential of interaction between convolution operations and further promotes the fusion of local features. The non-local spatial-spectral attention module is designed to capture the non-local spatial-spectral correlation in the hyperspectral image. Specifically, this module calculates the channel attention matrix to capture the global correlations between all of the spectral channels, and it fuses the channel attention attained feature maps and the spatial attention weighted features as the module output, thus both of the spatial-spectral correlations of hyperspectral images can be fully utilized for reconstruction. In addition, a compound loss, including the reconstruction loss, the measurement loss, and the cosine loss, is designed to guide the end-to-end network learning. We experimentally evaluate the proposed method on simulation and real datasets. The experimental results show that the proposed network outperforms the competing methods in terms of the reconstruction quality and running time.