We consider improving the polyhedral representation of the extensive form of a stochastic mixed-integer program (SMIP). Given a facet for a single-scenario version of an SMIP, our main result provides necessary and sufficient conditions under which this inequality remains facet-defining for the extensive form. We then present several implications, which show that common recourse structures from the literature satisfy these conditions. For example, for an SMIP with simple recourse, any single-scenario facet is also a facet for the extensive form. For more general recourse structures, we provide additional mild necessary and sufficient conditions.