  Reported here are transverse single-spin asymmetries ($A_{N}$) in the
production of charged hadrons as a function of transverse momentum ($p_T$) and
Feynman-$x$ ($x_F$) in polarized $p^{\uparrow}$+$p$, $p^{\uparrow}$+Al, and
$p^{\uparrow}$+Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV. The measurements
have been performed at forward and backward rapidity ($1.4<|\eta|<2.4$) over
the range of $1.5<p_{T}<7.0~{\rm GeV}/c$ and $0.04<|x_{F}|<0.2$. A nonzero
asymmetry is observed for positively charged hadrons at forward rapidity
($x_F>0$) in $p^{\uparrow}$+$p$ collisions, whereas the $p^{\uparrow}$+Al and
$p^{\uparrow}$+Au results show smaller asymmetries. This finding provides new
opportunities to investigate the origin of transverse single-spin asymmetries
and a tool to study nuclear effects in $p$+$A$ collisions.
