We demonstrate experimentally and computationally an intricate cavity size dependence of the anomalous near-infrared mode spectrum of an ordinary optical resonator that is combined with a ZnO:Ga-based hyperbolic metamaterial (HMM). Specifically, we reveal the existence of a resonance in subwavelength-sized cavities and demonstrate control over the first-order cavity mode dispersion. We elaborate that these effects arise due to the HMM combining the mode dispersions of purely metallic and purely dielectric cavity cores into a distinct intermediate regime. By tailoring the HMM fill factor, this unique dispersion of a subwavelength resonator can be freely tuned between these two limiting cases.