  Due to the lack of free neutron targets, studies of the structure of the
neutron are typically made by scattering electrons from either $^2$H or $^3$He
targets. In order to extract useful neutron information from a $^3$He target,
one must understand how the neutron in a $^3$He system differs from a free
neutron by taking into account nuclear effects such as final state interactions
and meson exchange currents. The target single spin asymmetry $A_y^0$ is an
ideal probe of such effects, as any deviation from zero indicates effects
beyond plane wave impulse approximation. New measurements of the target single
spin asymmetry $A_y^0$ at $Q^2$ of 0.46 and 0.96 (GeV/$c)^2$ were made at
Jefferson Lab using the quasi-elastic $^3\mathrm{He}^{\uparrow}(e,e'n)$
reaction. Our measured asymmetry decreases rapidly, from $>20\%$ at $Q^2=0.46$
(GeV/$c)^2$ to nearly zero at $Q^2=0.96$ (GeV$/c)^2$, demonstrating the
fall-off of the reaction mechanism effects as $Q^2$ increases. We also observed
a small $\epsilon$-dependent increase in $A_y^0$ compared to previous
measurements, particularly at moderate $Q^2$. This indicates that upcoming high
$Q^2$ measurements from the Jefferson Lab 12 GeV program can cleanly probe
neutron structure from polarized $^3$He using plane wave impulse approximation.
