Microphone Amplification
At least for now, just add a small amplifier after the one you already have.  A gain of 20, with maybe a volume control in front of it should give you the range you need with some margin.

You can rig up a little amplifier like that with something like a TL072 opamp and &plusmn;12 V supply.

Once you have your system working, and therefore know what you really need, then you can look at streamlining the whole amplifier chain.

Keep in mind that is not mostly about gain, but about signal to noise ratio.  Amplifying a noisy signal does nothing useful.