Finding the shortest paths for packets from sources to destinations in packet-switched communication networks is an inevitable problem in building a future high-speed information society. A routing method with memory information has already been proposed to alleviate the congestion of large volumes of packet flow. This routing method shows a high transmission completion rate even for large volumes of packet flows in communication networks with scale-free properties. However, the method exhibits poor performance for networks with local triangular connections and long distances between nodes. To overcome these problems, in this study, we first enhanced the routing performance of the conventional communication network models by using the betweenness centrality of nodes, which is one of the network centralities that measures the number of shortest paths that pass through each node in the networks. Subsequently, we adaptively changed the transmitting paths of packets by using only local information. Numerical simulations indicated that our routing method performs successfully for various topologies of communication networks by avoiding the congested node, and effectively using the memory information.