A simple "Baker clamp" as suggested by Unclejed should improve things.
Just add a schottky diode between base and collector.
That prevents saturation ever occurring by diverting the base current when the collector-emitter voltage reaches around 0.2V
ps. Q2 in the schematic you posted has no turn-on current source for its base. When Q1 turns off, there is nothing that biases the base above the emitter. Try adding a resistor from the collector of Q1 to power.