Through the use of particle-in-cell simulations, we study the ion flux asymmetry in an argon discharge that is induced by a ?sawtooth-like? excitation voltage waveform. In a previous article we have shown that, due to their differing rising and falling slopes, these waveforms can create a plasma with a significantly higher ion flux to one electrode in a geometrically symmetric reactor. Furthermore, they have the unique property of providing a lower ion energy at the electrode with a higher ion flux. In the present work, we show that a refined waveform allows the ion flux asymmetry to be increased for a given number of harmonics by reducing the ionization rate in front of the low-flux electrode. The flux asymmetry is found to disappear at low pressure due to the increased electron energy transport, which causes a transition from sheath edge ionization to bulk ionization. Changing the fundamental frequency is shown to have two counterbalancing effects: reducing the ionization on the low ion-flux electrode and shifting the maximum ionization to the center of the discharge. Under the representative conditions that we have studied, a maximum asymmetry is found for a base frequency of 3.4 MHz. Finally, it is shown that, by adjusting the rise- to fall-time ratio of the refined waveforms, the ion-flux asymmetry can be continuously shifted from one electrode to the other.