  A significant fraction (30\%) of well-localized short gamma-ray bursts
(sGRBs) lack a coincident host galaxy. This leads to two main scenarios:
\textit{i}) that the progenitor system merged outside of the visible light of
its host, or \textit{ii}) that the sGRB resided within a faint and distant
galaxy that was not detected by follow-up observations. Discriminating between
these scenarios has important implications for constraining the formation
channels of neutron star mergers, the rate and environments of gravitational
wave sources, and the production of heavy elements in the Universe. In this
work, we present the results of our observing campaign targeted at 31 sGRBs
that lack a putative host galaxy. Our study effectively doubles the sample of
well-studied sGRB host galaxies, now totaling 72 events of which $28\%$ lack a
coincident host to deep limits ($r$\,$\gtrsim$\,$26$ or
$F110W$\,$\gtrsim$\,$27$ AB mag), and represents the largest homogeneously
selected catalog of sGRB offsets to date. We find that 70\% of sub-arcsecond
localized sGRBs occur within 10 kpc of their host's nucleus, with a median
projected physical offset of $5.6$ kpc. Using this larger population, we
discover an apparent redshift evolution in their locations: bursts at low-$z$
occur at $2\times$ larger offsets compared to those at $z$\,$>$\,$0.5$. This
evolution could be due to a physical evolution of the host galaxies themselves
or a bias against faint high-$z$ galaxies. Furthermore, we discover a sample of
hostless sGRBs at $z$\,$\gtrsim$\,$1$ that are indicative of a larger high-$z$
population, constraining the redshift distribution and disfavoring log-normal
delay time models.
