The transverse polarization of $\Lambda$ hyperons within unpolarized jets originates from the transverse-momentum-dependent (TMD) fragmentation function $D_{1T}^\perp (z, p_T, \mu^2)$. In the vacuum environment, the QCD evolution of this TMD fragmentation function is governed by the Collins-Soper equation. However, in the presence of the quark-gluon plasma (QGP) medium, the jet-medium interaction induces a transverse-momentum-broadening effect that modifies the QCD evolution. As a result, the transverse spin polarization of $\Lambda$ hyperons in relativistic heavy-ion collisions differs from that in $pp$ collisions. We demonstrate that this difference serves as a sensitive probe for studying jet-medium interaction, offering a novel perspective through the spin degree of freedom.