  Fully understanding the average core-collapse supernova requires detecting
the diffuse supernova neutrino background (DSNB) in all flavors. While the DSNB
$\bar{\nu}_e$ flux is near detection, and the DSNB $\nu_e$ flux has a good
upper limit and prospects for improvement, the DSNB $\nu_x$ (each of $\nu_\mu,
\nu_\tau, \bar{\nu}_\mu, \bar{\nu}_\tau$) flux has a poor limit and heretofore
had no clear path for improved sensitivity. We show that a succession of
xenon-based dark matter detectors -- XENON1T (completed), XENONnT/LUX-ZEPLIN
(running), and DARWIN (proposed) -- can dramatically improve sensitivity to
DSNB $\nu_x$ the neutrino-nucleus coherent scattering channel. XENON1T could
match the present sensitivity of $\sim 10^3 \;
\mathrm{cm}^{-2}~\mathrm{s}^{-1}$ per $\nu_x$ flavor, XENONnT/LUX-ZEPLIN would
have linear improvement of sensitivity with exposure, and a long run of DARWIN
could reach a flux sensitivity of $\sim 10 \;
\mathrm{cm}^{-2}~\mathrm{s}^{-1}$. Together, these would also contribute to
greatly improve bounds on non-standard scenarios. Ultimately, to reach the
standard flux range of $\sim 1 \; \mathrm{cm}^{-2}~\mathrm{s}^{-1}$, even
larger exposures will be needed, which we show may be possible with the series
of proposed lead-based RES-NOVA detectors.
