Threefold-symmetric (TFS) Bricard linkages are known for their excellent deployment performance properties. This paper proposes a novel networking method of TFS Bricard linkages and a double-layer mechanical network. First, the angle relationship for parts of the TFS Bricard linkage is analyzed. Then, the angle relationship of two TFS Bricard linkages connected by a scissor mechanism is studied. The result suggests that when the twist angles of the two TFS Bricard linkages are equal, their corresponding planes are parallel, and the link lengths have no effect on the parallel relationship. A novel networking method of the TFS Bricard linkage is recommended according to these results. This mechanical network is constructed of two different sized units and can be plane deployed and be folded with a smaller height. We also propose a hybrid linkage constructed of the TFS Bricard linkage and Sarrus linkage. Two kinds of double-layer mechanical networks are suggested by applying the hybrid linkage to a smaller unit in the mechanical network and using the hybrid linkage as the interlayer pillar. The new networking method and the double-layer mechanical network provide convenience for the TFS Bricard linkage's engineering application.