Using $e^+e^-toLambda_c^+barLambda_c^-$ 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 $Sigma^0pi^+$. 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 $pK_S^0$, $Lambdapi^+$, $Sigma^+pi^0$ and $Sigma^0pi^+$ are measured to be $0.18pm0.43(rm{stat})pm0.14(rm{syst})$, $-0.80pm0.11(rm{stat})pm0.02(rm{syst})$, $-0.57pm0.10(rm{stat})pm0.07(rm{syst})$, and $-0.73pm0.17(rm{stat})pm0.07(rm{syst})$, respectively. In comparison with previous results, the measurements for the $Lambdapi^+$ and $Sigma^+pi^0$ modes are consistent but with improved precision, while the parameters for the $pK_S^0$ and $Sigma^0pi^+$ modes are measured for the first time.