We present a novel postprocessing technique for a discontinuous Galerkin (DG) discretization of time-dependent Maxwell's equations that we couple with an explicit Runge-Kutta time-marching scheme. The postprocessed electromagnetic field converges one order faster than the unprocessed solution in the H(curl)-norm. The proposed approach is local, in the sense that the enhanced solution is computed independently in each cell of the computational mesh, and at each time step of interest. As a result, it is inexpensive to compute, especially if the region of interest is localized, either in time or space. The key ideas behind this postprocessing technique stem from hybridizable discontinuous Galerkin (HDG) methods , which are equivalent to the analyzed DG scheme for specific choices of penalization parameters. We present several numerical experiments that highlight the superconvergence properties of the postprocessed electromagnetic field approximation.