So there's a separate equation then for audio power?
There is a graph in the datasheet for the LM386 that shows output p-p voltage swing vs supply voltage. Into an 8 ohm speaker the swing with a 5V supply is 3V and with a 9V supply the swing is 6V.
An equation for output power is RMS voltage swing squared divided by the speaker impedance. Then with a 5V supply the output is 3V divided by (2 x the root of 2)= 1.06V RMS and for a 9V supply the output is 6V (2 x the root of 2)= 2.12V RMS. You can calculate the current then calculate the power or you can use the equation V RMS squared divided by the speaker impedance. Then with a 5V supply the power is 0.14W and for a 9V supply the power is 0.56W.
So my easiest solution here is to raise the voltage of the LM386's VS pin for more audio output wattage? If so, what should I use for minimum voltage? I'll be running my circuit on batteries alongside other circuits running through a 7805 voltage regulator.
The graph of output p-p voltage swing vs supply voltage on the datasheet for the LM386 shows that increasing the supply above about 10V does not increase the output power. Another graph shows that the dissipation (heating) increases instead.
The minimum input for a 7805 regulator is 7V but a 9V battery drops to 6V or less during its life so use a "low dropout" 5V regulator that works fine when its input is as low as 5.5V.
I think you should look at Texas Instruments list of modern audio power amplifier ICs and select one that uses class-D (switching instead of linear for low heating) and has a bridged output (drives both wires of the speaker) for much more output power.
A single transistor is a heater not an amplifier. It operates in "class-A" (look in Google) and 3/4 of the battery power produces heat when it has maximum undistorted output. Its heating continues wasting battery power even when there is no input signal.
A linear amplifier like the LM386 and Harry's amplifier operates in class-AB and its heating is about 4/10ths to half of the battery power but almost no heating when there is no input signal.
A class-D amplifier is always efficient and cool.