  It has been recently proposed that the boosted dark matter (BDM) by supernova
neutrinos (SN$\nu$) from SN1987a or from the next galactic SN can serve as a
novel component to probe nonvanishing interaction between DM and the Standard
Model leptons. In this work, we extend this concept and evaluate the
present-day diffuse flux of SN$\nu$ BDM originated from all galaxies at higher
redshifts. We show that by considering this diffuse BDM (DBDM) component, the
best model-independent sensitivity on the product of the DM-$\nu$ and
DM-electron cross sections, $\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}\simeq
\mathcal{O}(10^{-37})$~cm$^2$ for sub-MeV DM, can be obtained with large-size
neutrino experiments such as Super-Kamiokande or Hyper-Kamiokande, surpassing
the estimated SN$\nu$ BDM bound from SN1987a. We also examine the impact due to
the presence of DM spikes around the supermassive black holes in galaxies on
SN$\nu$ BDM and DBDM. Our results suggest that both the DBDM and the SN$\nu$
BDM probes are insensitive to the uncertain properties of DM spikes, unless the
next galactic SN happens to occur at a location extremely close to or right
behind the galactic center along the SN line of sight.
