Hybrid turnstiles have proven to generate accurate single-electron currents that underpin electric current metrology. The usual turnstile operation involves setting the working point of the device by adjusting the DC bias and gate voltages, and application of an AC harmonic signal to the gate. This corresponds to the straight pumping trajectory on the bias voltage–gate voltage plane and results in the well-defined current plateaus in the transport characteristics. The accuracy of charge pumping at high operation frequencies is limited, among other errors, by the so-called back-tunnelling events when electrons tunnel in the direction against the applied bias. Increase of the barrier transparency between the island and the leads as well as the source-drain bias suppresses these, but leads to some additional errors. We experimentally demonstrate a driving protocol that suppresses back-tunnelling, thus extending the range of frequencies for generating accurate single-electron currents. The main feature of this approach is an additional AC harmonic signal applied to the bias with frequency twice that applied to the gate electrode. This allows additional modulation of the island chemical potential and improves the single-electron current accuracy by one order of magnitude. We also show that accurate single-electron currents can be generated at zero average bias voltage by applying only AC drive to the bias. Furthermore, we show that within this protocol, the current direction can be reversed by shifting the phase of the source signal. This novel drive method can find applications in quantum sensing and quantum computing.