  We perform a high-resolution cosmological zoom-in simulation of a Milky Way
(MW)--like system, which includes a realistic Large Magellanic Cloud analog,
using a large differential elastic dark matter self-interaction cross section
that reaches $\approx 100~\mathrm{cm}^2\ \mathrm{g}^{-1}$ at relative
velocities of $\approx 10~\mathrm{km\ s}^{-1}$, motivated by the diverse and
orbitally dependent central densities of dwarf galaxies within and surrounding
the MW. We explore the effects of dark matter self-interactions on satellite,
splashback, and isolated halos through their abundance, central densities,
maximum circular velocities, orbital parameters, and correlations between these
variables. We use an effective constant cross section model to analytically
predict the stages of our simulated halos' gravothermal evolution,
demonstrating that deviations from the collisionless $R_{\rm max}$--$V_{\rm
max}$ relation can be used to select deeply core-collapsed halos, where $V_{\rm
max}$ is a halo's maximum circular velocity, and $R_{\rm max}$ is the radius at
which it occurs. We predict that a sizable fraction ($\approx 20\%$) of
subhalos with masses down to $\approx 10^8~M_{\odot}$ is deeply core collapsed
in our SIDM model. Core-collapsed systems form $\approx 10\%$ of the isolated
halo population down to the same mass; these isolated, core-collapsed halos
would host faint dwarf field galaxies with extremely steep central density
profiles. Finally, most halos with masses above $\approx 10^9~M_{\odot}$ are
core-forming in our simulation. Our study thus demonstrates how
self-interactions diversify halo populations in an environmentally dependent
fashion within and surrounding MW-mass hosts, providing a compelling avenue to
address the diverse dark matter distributions of observed dwarf galaxies.
