We use scanning near-field optical microscopy to study the response of hexagonal boron nitride nanocones at infrared frequencies, where this material behaves as a hyperbolic medium. The obtained images are dominated by a series of "hot" rings that occur on the sloped sidewalls of the nanocones. The ring positions depend on the incident laser frequency and the nanocone shape. Both dependences are consistent with directional propagation of hyperbolic phonon-polariton rays that are launched at the edges and zigzag through the interior of the nanocones, sustaining multiple internal reflections off the sidewalls. Additionally, we observe a strong overall enhancement of the near-field signal at discrete resonance frequencies. These resonances attest to low dielectric losses that permit coherent standing waves of the subdiffractional polaritons to form. We comment on potential applications of such shape-dependent resonances and the field concentration at the hot rings.