Assume all rings have 1, preserved by homomorphisms, and that all modules are unitary.
Given a ring homomorphism $\phi:R\to S$, any $S$-module A can be given an $R$-module structure by the action $ra\mapsto\phi(r)a$.
Given again the ring homomorphism $\phi$, what other conditions are needed to utilize $\phi$ for giving any $R$-module an $S$-module structure?
For the ring homomorphism $\phi: R \to S$ you have described the functor of “restriction along $\phi$”, which takes an $S$-module and views it as an $R$ module by using $\phi$. There is a functor in the opposite direction, called “induction along $\phi$”, which takes any $R$-module and produces an $S$-module.
Induction is slightly trickier to explain. We can view $S$ as an $(S, R)$ bimodule, via the actions $s \cdot x = sx$ and $x \cdot r= x \phi(r)$. Then, if $V$ is a left $R$-module, the tensor product $$ S \otimes_R V$$ is the “induction of $V$ along $\phi$”.