The wireless Internet-access mesh network (WIMNET) can realize an inexpensive flexible Internet-access network by using multiple access points (APs) connected by wireless links. For WIMNET with a single Internet Gateway (GW), we have proposed the active AP selection algorithm of selecting active APs from redundantly allocated APs for communications. However, in practical implementations of WIMNET, multiple GWs usually exist and the number of hops between a GW and a host is limited. Furthermore, the communication speed is dynamically changed by the link distance for the new high-speed communication protocol IEEE802.11ac. In this paper, we propose an extension of the active AP selection algorithm to consider these conditions. We verify the effectiveness of our proposal through simulations in four instances using the WIMNET simulator.