How can I modify this circuit for a higher gain?
There is another way to set the gain, which the datasheet briefly mentions but it doesn't go into any detail.

"Gain control can also be done by capacitively coupling  a resistor (or FET) from pin 1 to ground."

See:

http://www.wentztech.com/radio/resources/Technical-Info/_LM386/Unleash-lm386.pdf

which discusses how to use that method to get a gain of over 4500 (73dB).  It requires careful layout and grounding.