For high-precision cosmic-ray measurements using the radio emission of air showers such as done with LOFAR and envisioned for the Square Kilometre Array (SKA), simulating ensembles of air showers takes considerable CPU time. In particular, simulating every antenna position would become intractable for the 60,000 antennas at SKA. We present a higher-order algorithm for interpolating radio signals of air showers simulated on a radial (polar) grid. It allows to interpolate the full pulse signal in each antenna/polarization along any position in the footprint. It provides significant improvement over existing, linear methods. From a set of simulated showers analyzed from 30 to 500 MHz, we show that simulating about 200 antennas is sufficient to represent signals at any relevant position to high accuracy. Thus, by reducing simulation time for SKA by a factor ∼ 300, it offers a relevant extension to radio simulation codes.