  We explore cosmological magnetogenesis in the post-inflationary universe,
when the inflaton oscillates around its potential minimum and the universe is
effectively dominated by cold matter. During this epoch prior to reheating,
large-scale magnetic fields can be significantly produced by the cosmological
background. By considering magnetogenesis both during and after inflation, we
demonstrate that magnetic fields stronger than 10^{-15} G can be generated on
Mpc scales without having strong couplings in the theory, or producing too
large electric fields that would dominate the universe.
