The inherent crystal anisotropy of hexagonal boron nitride (hBN) provides the ability to support hyperbolic phonon polaritons, that is, polaritons that can propagate with very large wave vectors within the material volume, thereby enabling optical confinement to exceedingly small dimensions. Indeed, previous research has shown that nanometer-scale truncated nanocone hBN cavities, with deep subdiffractional dimensions, support three-dimensionally confined optical modes in the mid-infrared. Because of optical selection rules, only a few of the many theoretically predicted modes have been observed experimentally via far-field reflection and scattering-type scanning near-field optical microscopy (s-SNOM). The photothermal induced resonance (PTIR) technique probes optical and vibrational resonances overcoming weak far-field emission by leveraging an atomic force microscope (AFM) probe to transduce local sample expansion caused by light absorption. Here we show that PTIR enables the direct observation of previously unobserved, dark hyperbolic modes of hBN nanostructures. Leveraging these optical modes and their wide range of angular and radial momenta could provide a new degree of control over the electromagnetic near-field concentration, polarization in nanophotonic applications.