By introducing an additional air hole into the center of each unit cell of the conventional honeycomb lattice, a new kind of structure that can support air guidance of light with low-order bandgaps is proposed. Numerical investigation indicates that, due to the generation of the lower order bandgaps, larger air-guiding photonic bandgaps (PBGs) than those of triangular structures can be achieved in the new structures. Furthermore, the proposed structures also provide additional freedom in tailoring PBGs.