The scan rate of an axion haloscope is proportional to the square of the cavity volume. In this paper, a new class of thin-shell cavities are proposed to search for axionic dark matter. These cavities feature active volume much larger (>20X) than that of a conventional cylindrical haloscope, comparable quality factor Q, and a similar frequency tuning range. Full 3D numerical finite-element analyses have been used to show that the TM_010 eigenmodes are singly polarized throughout the volume of the cavity and can facilitate axion-photon conversion in uniform magnetic field produced by a superconducting solenoid. To mitigate spurious mode crowding and volume loss due to localization, a pre-amplification binary summing network will be used for coupling. Because of the favorable frequency-scaling, the new cavities are most suitable for centimeter-wavelength (~ 10-100 GHz), corresponding to the promising post-inflation axion production window. In this frequency range, the tight machining tolerances required for high-Q thin-shell cavities are achievable with standard machining techniques for near-infrared mirrors.