Connected-annular-rods photonic crystals (CARPCs) in both triangular and square lattices are proposed to enhance the two-dimensional complete photonic bandgap (CPBG) for chalcogenide material systems with moderate refractive index contrast. For the typical chalcogenide-glass–air system with an index contrast of 2.8:1, the optimized square lattice CARPC exhibits a significantly larger normalized CPBG of about 13.50%, though the use of triangular lattice CARPC is unable to enhance the CPBG. It is almost twice as large as our previously reported result [IEEE J. Sel. Top. Quantum Electron.22, 4900108 (2016)IJSQEN1077-260X10.1109/JSTQE.2015.2422997]. Moreover, the CPBG of the square-lattice CARPC could remain until an index contrast as low as 2.24:1. The result not only favors wideband CPBG applications for index contrast systems near 2.8:1, but also makes various optical applications that are dependent on CPBG possible for more widely refractive index contrast systems.