Public safety networks need to more effectively meet the increasing demands for images or videos to be shared among first responders and incident commanders. Long term evolution (LTE) networks are considered to be candidates to achieve such broadband services. Capital expenditures in deploying base stations need to be decreased to introduce LTE for public safety. However, out-of-coverage areas tend to occur in cell edge areas or inside buildings because the cell areas of base stations for public safety networks are larger than those for commercial networks. The 3rd Generation Partnership Program (3GPP) in Release 13 has investigated device-to-device (D2D) based relay communication as a means to fill out-of-coverage areas in public safety LTE (PS-LTE). This paper proposes a relay selection scheme based on effective path throughput from an out-of-coverage terminal to a base station via an in-coverage relay terminal, which enables the optimal relay terminal to be selected. System level simulation results assuming on radii of 20km or less revealed that the proposed scheme could provide better user ratios that satisfied the throughput requirements for video transmission than the scheme standardized in 3GPP. Additionally, an evaluation that replicates actual group of fire-fighters indicated that the proposed scheme enabled 90% of out-of-coverage users to achieve the required throughput, i.e., 1.0Mbps, to transmit video images.