The rapid spread of smart phones causes the explosion of media traffic. This encourages mobile network operators (MNOs) to coordinate multiple access networks. For such an MNO, the IP Multimedia Subsystem (IMS) is a promising service control infrastructure to ensure a QoS guaranteed communication path for the media in the multi-access network. IMS-based service continuity enables user equipments (UEs) to continuously use IMS-based services (e.g., VoIP) even when the UEs make handovers between different access networks where the UE is assigned the different IP address, respectively. In the case where the UE cannot simultaneously use multiple wireless devices, there is the possibility of a long media disruption time during handovers. This is caused by several consecutive handovers as a result of attempting to discover the access network where the UE can have the QoS-guaranteed communications. In this paper, we propose a method for reducing the media disruption time when the UE makes handovers between different access networks. In the proposal, the UE proactively performs the service continuity procedure, and selects an access network that can provide the required network resources to the UE. We implement and evaluate the proposed method, and show how the media disruption time can be reduced.