Multidimensional imaging has emerged as a powerful technology capable of simultaneously acquiring spatial, spectral, and depth information about a scene. However, existing approaches often rely on mechanical scanning or multi-modal sensing configurations, leading to prolonged acquisition times and increased system complexity. Coded aperture snapshot spectral imaging (CASSI) has introduced compressed sensing to recover three-dimensional (3D) spatial-spectral datacubes from single snapshot two-dimensional (2D) measurements. Despite its advantages, the reconstruction problem remains severely underdetermined due to the high compression ratio, resulting in limited spatial and spectral reconstruction quality. To overcome this challenge, we developed a novel two-stage cascaded compressed sensing scheme called coded aperture snapshot hyperspectral light field tomography (CASH-LIFT). By appropriately distributing the computation load to each stage, this method utilizes the compressibility of natural scenes in multiple domains, reducing the ill-posed nature of datacube recovery and achieving enhanced spatial resolution, suppressed aliasing artifacts, and improved spectral fidelity. Additionally, leveraging the snapshot 3D imaging capability of LIFT, our approach efficiently records a five-dimensional (5D) plenoptic function in a single snapshot.