In-band full-duplex communication offers significant potential to enhance network performance. This paper presents the full-duplex linear transmit delay allocation MAC (FD-LTDA-MAC) protocol for full-duplex based underwater acoustic chain networks (FD-UACNs) for subsea pipeline monitoring. This incorporates a number of extensions to the LTDA-MAC protocol in order to fully exploit advantages of full-duplex communication to enhance the efficiency of underwater facility monitoring. The protocol uses a greedy optimisation algorithm to derive collision-free packet schedules for delivering data packets to the sink node of the underwater chain network. The purpose of this paper is to show the significant improvement that can be achieved in packet scheduling by exploiting temporal spectrum re-use of an underwater acoustic channel through full-duplex communication. Simulation results show that more efficient packet scheduling and reduced end-to-end packet delays can be achieved in large scale scenarios using FD-LTDA-MAC compared with LTDA-MAC and LTDA-MAC with full-duplex enabled nodes. It can provide much higher monitoring rates for long range underwater pipelines using low cost, mid range, low rate, and low power acoustic modems.