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We present a solution method which combines the method of matched asymptotics with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonators arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the specially-scaled limit). In this regime, the existing treatment in Part I, with updated parameters, is found to return spurious spectral behaviour. We derive a regularised system which overcomes this issue and also derive compact asymptotic descriptions for the low-frequency dispersion equation in this setting. In the specially-scaled limit, our asymptotic dispersion equation not only recovers the first band surface but also extends to high, but still subwavelength, frequencies. A homogenisation treatment is outlined for describing the effective bulk modulus and effective density tensor of the resonator array for all wall thicknesses. We demonstrate that specially-scaled resonators are able to achieve exceptionally low Helmholtz resonant frequencies, and present closed-form expressions for determining these explicitly. We anticipate that the analytical expressions and the formulation outlined here may prove useful in industrial and other applications.