I should caveat the xpl2 statement. You will only get more output if you drive it over about 5.5 amps. It will require less voltage. So it will be more efficient. Below is a chart made by Texas_Ace in this thread.
Regarding 6V verses 4.2 volts. Four 4 alkaline cells can not deliver 6 volts under load due to significant voltage sag at high currents. Li-ion will also have voltage sage, but significantly less depending on the cell. Here are some AAA discharge curves. The bottom red line is only a .5 amp load. Chart from this site:
I don't know anything about your stock driver. I thought I read on CPF that the LP600 was direct drive, but I may have misread that. Can you post a picture of the top and bottom of the driver? There are some sweet Direct Drive drivers available, but I'm a bit out of touch were to get them right now.