> What can I do to improve the volume?  I have heard the same beeper
> in other devices generate louder beeps, so there must be room for
> improvement on my side.

Drive it at it's resonant frequency.
They are VERY "peaky", if you drive them at some other rate, they are rather
quiet.

Using three inverters, you can drive one side with one inverter, and the
other side with two in series, which effectively doubles the applied
voltage.