  Next-generation large-volume detectors, such as GRAND, POEMMA, Trinity,
TAROGE-M, TAMBO, or PUEO, have been designed to search for ultra-high-energy
cosmic rays (UHECRs) with unprecedented sensitivity. We propose to use these
detectors to search for new physics beyond the Standard Model (BSM). By
considering the simple case of a right-handed neutrino that mixes exclusively
with the active $\tau$ neutrino, we demonstrate that the existence of new
physics can increase the probability for UHECRs to propagate through the Earth
and produce extensive air showers that will be measurable soon. We compare the
fluxes of such showers that would arise from various diffuse and transient
sources of high-energy neutrinos, both in the Standard Model and in the
presence of a right-handed neutrino. We show that detecting events with
emergence angles $\gtrsim 10$ deg is promising to probe the existence of BSM
physics, and we study the sensitivity of GRAND and POEMMA to do so. In
particular, we show that the hypothesis of a right-handed neutrino with a mass
of $\mathcal O(1-16)$ GeV may be probed in the future for mixing angles as
small as $|U_{\tau N}|^2 \gtrsim 10^{-7}$, thus competing with existing and
projected experimental limits.
