We present frequency-tuning mechanisms for dielectric resonators, which undergo "supermode" interactions as they tune. The tunable schemes are based on dielectric materials strategically placed inside traditional cylindrical resonant cavities, necessarily operating in transverse-magnetic modes for use in axion haloscopes. The first technique is based on multiple dielectric disks with radii smaller than that of the cavity. The second scheme relies on hollow dielectric cylinders similar to a Bragg resonator, but with a different location and dimension. Specifically, we engineer a significant increase in form factor for the TM030 mode utilizing a variation of a distributed Bragg reflector resonator. Additionally, we demonstrate an application of traditional distributed Bragg reflectors in TM modes which may be applied to a haloscope. Theoretical and experimental results are presented showing an increase in Q factor and tunability due to the supermode effect. The TM030 ring-resonator mode offers a between 1 and 2-order-of-magnitude improvement in axion sensitivity over current conventional cavity systems and will be employed in the forthcoming ORGAN experiment.