An efficient flow assignment strategy is of great importance to alleviate traffic congestion on multilayer networks. In this work, by considering the roles of nodes' local structures on the microlevel, and the different transporting speeds of layers in the macrolevel, an effective traffic-flow assignment strategy on multilayer networks is proposed. Both numerical and semianalytical results indicate that our proposed flow assignment strategy can reasonably redistribute the traffic flow of the low-speed layer to the high-speed layer. In particular, preferentially transporting the packets through small-degree nodes on the high-speed layer can enhance the traffic capacity of multilayer networks. We also find that the traffic capacity of multilayer networks can be improved by increasing the network size and the average degree of the high-speed layer. For a given multilayer network, there is a combination of optimal macrolevel parameter and optimal microlevel parameter with which the traffic capacity can be maximized. It is verified that real-world network topology does not invalidate the results. The semianalytical predictions agree with the numerical simulations.