If you need more power - assumedly so that you can drive your speakers louder - get a different receiver (or integrated amp, or preamp+power amp), OR, get more sensitive speakers.
And keep in mind that it takes twice as much power to get a 3dB increase in loudness (3dB is generally defined as the smallest change in volume to be clearly noticeable...a 10dB change = twice as loud, and requires 8 times as much power).
Your 680, like any stereo amp, has two amps - one for the right channel, and one for the left. In the 680 (and most receivers), the power supply for the amps is shared. Each of these amps is rated to be able to deliver 30 watts of continuous output into an 8 ohm load.
Whether you use two receivers or one, the power available for each speaker is therefore the same....30 watts, assuming 8 ohm speakers (slightly higher with lower impedance speakers).
While true that by using two receivers the common power supply is not shared, this will not provide an appreciable difference in output.