Maintaining a high overall network reliability remains one of the most critical requirements for advanced metering infrastructures (AMIs) in smart grid. Ensuring reliable networks not only determines the robust communications of an AMI, but also guarantees assured information delivery in the access network. To prevent any communication failures, incremental designs based on legacy networks should be carried out in advance to improve the overall redundancy. Current communication architecture of an AMI follows a traditional access network structure with a tree-based topology, which does not always satisfy high robustness and is prone to network failures. To address the challenge, this study conducts a reliability study of the access network in an AMI. Specifically, this study first examines the basic network topology adopted in an AMI access network and its underlying connectivity issues. Secondly, this study proposes two practical solutions as parts of incremental network design to improve the communication robustness of existing communication architectures. Thirdly, mathematical models are formulated to solve network connectivity problems, for maintaining a high overall network reliability, while minimising the communication deployment cost at the same time. Simulation results are provided from the aspects of minimal path sets and minimal cut sets to demonstrate the redundancy analysis.