For the ill-posed tensor completion (TC) problem, how to appropriately explore the hidden information is a pivotal yet challenging issue. Two well-known priors have been intensively studied, including the shallow priors such as low-rankness and local piecewise smoothness, and the deep priors using data-driven learning. However, their singleton usage or the simple combination hardly touches an acceptable performance. In this paper, we propose a hybrid shallow-and-deep priors model (HPM) to simultaneously exploit the respective strengths for a better performance. Specifically, based on the frontiers, i.e., tensor nuclear norm of discrete cosine transform (DCTNN) and enhanced 3D total variation (E3DTV), we propose a weighting scheme to characterize both the globally low-rank correlation and the locally smoothness. On that basis, a data-driven denoiser following the Plug-and-Play (PnP) framework is incorporated to provide the implicit information that cannot be captured by the shallow priors. Finally, the overall tensor completion model is optimized by the well-known alternating direction method of multiplier (ADMM). Numerical experiments show that the hybrid priors benefit from both types of visual properties and enable state-of-the-art quantitative and qualitative performance.