Current and future searches for dark matter axions, based on their resonant conversion to photons in a magnetic field, span many orders of magnitude. A major impediment to designing resonators at the high end of this range, 5 GHz and above, is the proliferation of TE modes, which overwhelm and hybridize with the TM010 mode to which the axion couples, making the search impossible. We demonstrate that a photonic bandgap structure can be designed that completely suppresses the TE spectrum, even reducing the number of lattice periods to two or one and violating perfect lattice symmetry. This allows tunable resonators to be designed in a convenient, volumetrically efficient circular geometry thus enabling future searches in the post-inflation axion mass range.