Tensor-valued data benefit greatly from dimension reduction as the reduction in size is exponential in the number of modes. To achieve maximal reduction without loss of information, our objective in this work is to provide an automated procedure for the optimal selection of reduced dimensionality. Our approach combines a recently proposed data augmentation procedure with the higher-order singular value decomposition (HOSVD) in a tensorially natural way. We give theoretical guidelines on how to choose the tuning parameters and further inspect their influence in a simulation study. As our primary result, we show that the procedure consistently estimates the true latent dimensions under a noisy tensor model, both at the population and sample levels. Additionally, we propose a bootstrap-based alternative to the augmentation estimator. Simulations are used to demonstrate the estimation accuracy of the two methods under various settings. Supplementary materials for this article are available online.