By investigating a square-shaped metamaterial structure we discover that wave diffraction at diagonal corners of such a structure excites transverse magnetic harmonics of 210 mode (TM210 harmonics). Multi-layer overlapping and deliberately regulating period length between adjacent unit cells can significantly enhance TM210 harmonics, leading to a strong absorption waveband. On such a basis, a design strategy is proposed to achieve broadband, thin-thickness multi-layered metamaterial absorbers (MMAs). In this strategy big pyramidal arrays placed in the "white blanks" of a chessboard exhibit two isolated absorption bands due to their fundamental and TM210 harmonics, which are further connected by another absorption band from small pyramidal arrays in the "black blanks" of the chessboard. The as-designed MMA at a total thickness (h) of 4.36 mm shows an absorption of above 0.9 in the whole frequency range of 7-18 GHz, which is 38% broader with respect to previous design methods at the same h. This strategy provides an effective route to extend the absorption bandwidth of MMAs without increasing h.