Methods and apparatus for use by a mobile communication device in controlling the selection of communication networks for communications are described. In one illustrative example, the mobile device maintains a communication link with a radio access network (RAN) (e.g. an IEEE 802.11-based wireless network). The RAN is part of a generic access network (GAN) which further includes a gateway controller. The mobile device maintains, via the GAN, a service connection for a voice and/or data service with a core network of a visited Public Land Mobile Network (PLMN). The mobile device repeatedly performs actions while the service connection with the core network associated with the visited PLMN is maintained. Specifically, the mobile device operates to set and run a timer and, in response to an expiration of the timer, the mobile device sends, to a gateway controller via the RAN, a service connection request for a service connection with a core network of a home PLMN of the mobile device. If the service connection request is accepted, the mobile device operates to establish and maintain, via the GAN, a service connection with the core network of the home PLMN for communications. If this service connection request is not accepted, however, the mobile device operates to select an alternative visited PLMN from a preferred list of visited networks if the alternative visited PLMN has a higher priority in the list than the visited PLMN. In this case, the mobile device sends to a (gateway controller a service connection request for a service connection with a core network of the alternative visited PLMN.