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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.