Let $R$ be an associative ring with identity. We introduce the notion of semi-$\tau$-supplemented modules, which is adapted from srs-modules, for a preradical $\tau$ on $R$-Mod. We provide basic properties of these modules. In particular, we study the objects of $R$-Mod for $\tau=\operatorname{Rad}$. We show that the class of semi-$\tau$-supplemented modules is closed under finite sums and factor modules. We prove that, for an idempotent preradical $\tau$ on $R$-Mod, a module $M$ is semi-$\tau$-supplemented if and only if it is $\tau$-supplemented. For $\tau=\operatorname{Rad}$, over a local ring every left module is semi-$\operatorname{Rad}$-supplemented. We also prove that a commutative semilocal ring whose semi-$\operatorname{Rad}$-supplemented modules are a direct sum of $w$-local left modules is an artinian principal ideal ring.