The effects of a driven fast-oscillating spinneret on the bending instability of electrified jets, leading to the formation of spiral structures in electrospinning experiments with charged polymers, are explored by means of extensive computer simulations. It is found that the morphology of the spirals can be placed in direct correspondence with the oscillation frequency and amplitude. In particular, by increasing the oscillation amplitude and frequency, thinner fibers can be extracted by the same polymer material, thereby opening design scenarios in electrospinning experiments.