Underwater acoustic communications (UAC) features frequency-dependent signal attenuation, long propagation delay, and doubly-selective fading. Thus, the design of reliable and efficient UAC protocols is very challenging. On the other hand, cooperative relay communications, which can provide reliable data transmission, is a very attractive technology for UAC. In our recent work, we proposed a practical asynchronous relaying protocol tailored for UAC: Asynchronous Amplify-and-forward relaying with Precoded OFDM (AsAP). This protocol resolves both the time synchronization difficulty and the frequency-selective issue of UAC channels. However, the AsAP protocol adopts fixed amplification and uniform power allocation among the source and the relays, which limits the system performance. Thus, in this paper, we design a modified AsAP system with an instantaneous amplification scheme, and investigate the optimum power allocation based on the criteria of maximum average SNR at the destination. Assuming statistical channel state information (CSI) is available, we analyze the average SNR at the destination and perform the power optimization on the AsAP system. The analytical results suggest a selective relaying scheme, where only the relay with best average SNR forwards the source information. Therefore, we propose an AsAP protocol with selective relaying (SR-AsAP). Finally, the performance of the new SR-AsAP scheme is simulated, and the benefits are illustrated through comparison with the direct-link system and the AsAP protocol.