  The importance of detecting neutrinos from a Milky Way core-collapse
supernova is well known. An under-studied phase is proto-neutron star cooling.
For SN 1987A, this seemingly began at about 2 s, and is thus probed by only 6
of the 19 events (and only the $\bar{\nu}_e$ flavor) in the Kamiokande-II and
IMB detectors. With the higher statistics expected for present and near-future
detectors, it should be possible to measure detailed neutrino signals out to
very late times. We present the first comprehensive study of neutrino detection
during the proto-neutron star cooling phase, considering a variety of outcomes,
using all flavors, and employing detailed detector physics. For our nominal
model, the event yields (at 10 kpc) after 10 s -- the approximate duration of
the SN 1987A signal -- far exceed the entire SN 1987A yield, with $\simeq$250
$\bar{\nu}_e$ events (to 50 s) in Super-Kamiokande, $\simeq$110 $\nu_e$ events
(to 40 s) in DUNE, and $\simeq$10 $\nu_\mu, \nu_\tau, \bar{\nu}_\mu,
\bar{\nu}_\tau$ events (to 20 s) in JUNO. These data would allow unprecedented
probes of the proto-neutron star, including the onset of neutrino transparency
and hence its transition to a neutron star. If a black hole forms, even at very
late times, this can be clearly identified. But will the detectors fulfill
their potential for this perhaps once-ever opportunity for an all-flavor,
high-statistics detection of a core collapse? Maybe. Further work is urgently
needed, especially for DUNE to thoroughly investigate and improve its MeV
capabilities.
