We present a novel offline-online method to mitigate the computational burden of Bayesian inference, particularly in the regime where the posterior densities are computationally demanding to evaluate while real-time inference results are needed. In the offline phase, the proposed method learns the joint law of the parameter random variables and the observable random variables in the tensor-train (TT) format. Then, in the online phase, the resulting order-preserving transport can be conditioned on newly observed data to characterize the posterior random variables in real-time. Compared with the state-of-the-art normalizing flows techniques, our proposed method relies on function approximation, for which we can provide a thorough performance analysis. The function approximation perspective allows us to significantly improve the capability of transport maps in challenging problems with high-dimensional observations and high-dimensional parameters. Capitalizing on this, we present novel heuristics to either reorder or reparametrize the variables to enhance the approximation power of TT. We then integrate the TT-based transport maps and the parameter reordering/reparametrization into a layered composite map to further improve the performance of the resulting inference. We demonstrate the efficiency of the proposed method on various statistical learning tasks involving ordinary differential equations (ODEs) and partial differential equations (PDEs).