  Kinematics restrict the ability of rare charm decays to explore the charged
Lepton Flavor Violation processes mediated by the quark-level $c\to u \ell
\tau$ transition. To fill the gap, we propose exploring new physics (NP)
through the quasielastic scattering process $e^-p\to \tau^-\Lambda_c$ and the
polarization of the $\tau$ lepton. As analyzing modes for the $\tau$
polarization, we consider the decays $\tau^-\to \pi^-\nu_{\tau}$, $\tau^-\to
\rho^-\nu_{\tau}$, and $\tau^- \to \ell^-\bar{\nu}_{\ell}\nu_{\tau}$, and show
that the $\tau$ polarization components can be extracted from analyzing the
kinematics of the $\tau$ visible decay products. In the framework of a general
low-energy effective Lagrangian, we then perform a detailed analysis of the
polarization components in various aspects and scrutinize possible NP signals.
With one upcoming experimental setup, we finally demonstrate promising event
rate can be expected for the cascade process and, even in the worst-case
scenario -- no signals is observed at all -- it can still provide a competitive
potential for constraining the NP, compared with those from the high-$p_T$
dilepton invariant mass tails at high-energy colliders.
