  Using $e^+e^-\to\Lambda_c^+\bar\Lambda_c^-$ production from a 567 pb$^{-1}$
data sample collected by BESIII at 4.6 GeV, a full angular analysis is carried
out simultaneously on the four decay modes of $\Lambda_c^+\to pK_S^0$, $\Lambda
\pi^+$, $\Sigma^+\pi^0$, and $\Sigma^0\pi^+$. For the first time, the
$\Lambda_c^+$ transverse polarization is studied in unpolarized $e^+e^-$
collisions, where a non-zero effect is observed with a statistical significance
of 2.1$\sigma$. The decay asymmetry parameters of the $\Lambda_c^+$ weak
hadronic decays into $pK_S^0$, $\Lambda\pi^+$, $\Sigma^+\pi^0$ and
$\Sigma^0\pi^+$ are measured to be $0.18\pm0.43(\rm{stat})\pm0.14(\rm{syst})$,
$-0.80\pm0.11(\rm{stat})\pm0.02(\rm{syst})$,
$-0.57\pm0.10(\rm{stat})\pm0.07(\rm{syst})$, and
$-0.73\pm0.17(\rm{stat})\pm0.07(\rm{syst})$, respectively. In comparison with
previous results, the measurements for the $\Lambda\pi^+$ and $\Sigma^+\pi^0$
modes are consistent but with improved precision, while the parameters for the
$pK_S^0$ and $\Sigma^0\pi^+$ modes are measured for the first time.
