Are you talking power supplies or input signal voltages?
Just because the transmitter can accept voltages from 5-12v doesn't mean it will transmit with more power at 12v over 5.5V if you are talking power supply. It all depends on what the power is regulated down to at the transmitter. The input signal from the 3v board may fall right in line with min/max input signal voltages.
Using efficient antennas placed in ideal locations may be the best way to extend ranges.