  The tension between late and early universe probes of today's expansion rate,
the Hubble parameter $H_0$, remains a challenge for the standard model of
cosmology $\Lambda$CDM. There are many theoretical proposals to remove the
tension, with work still needed on that front. However, by looking at new
probes of the $H_0$ parameter one can get new insights that might ease the
tension. Here, we argue that neutrino oscillations could be such a probe. We
expand on previous work and study the full three-flavor neutrino oscillations
within the $\Lambda$CDM paradigm. We show how the oscillation probabilities
evolve differently with redshift for different values of $H_0$ and neutrino
mass hierarchies. We also point out how this affects neutrino fluxes which,
from their measurements at neutrino telescopes, would determine which value of
$H_0$ is probed by this technique, thus establishing the aforementioned aim.
