This paper studies a reformulation of a general spatially hybrid optimal control problem for which the dynamics is usually defined over a given partition of the state space into strata so that switchings of the system only occur regionally (i.e., at the time of a change of stratum). Given a global solution to such a prob- lem, we associate a temporally hybrid optimal control problem for which the change of dynamics now occurs at free switching instants that can be optimized (without considering a partition). We prove that under a strong transverse condition on the dynamics at the interfaces between strata, the global solution is a L1-local solution to this new problem. Thanks to an explicit example, we also prove that this reformu- lation fails to hold in general. In fact, the analysis of this example makes it possible to demonstrate that the structure of the solution to the spatially and temporally hybrid optimal control problems are different. The study carried out in this work highlights several ways for obtaining the spatially hybrid maximum principle.