We demonstrate using particle-in-cell simulations that electrons can be injected into a hybrid laser wakefield and direct laser accelerator via ionization injection. We propose an accelerator and injector scenario that utilizes two laser pulses. The first (pump) pulse produces the plasma ‘bubble’ by expelling the plasma electrons generated by its leading edge from the low-Z component of the gas mixture, and then injects electrons into the bubble by ionizing the high-Z component. The second time-delayed laser pulse resonantly interacts with these injected electrons undergoing betatron oscillations inside the bubble. We show that the electrons ionized off-axis and on-axis but off the peak ionization phase possess sufficient transverse energy to undergo efficient direct laser acceleration (DLA). When combined with their acceleration by the bubble’s longitudinal plasma wake, DLA can double the total energy gain and produce a monoenergetic beam.