I have a theory about what you're asking, but I'm not entirely sure. And I like to think in the simplest terms, so please don't be offended and think that I'm talking down or anything; I'm just seeking clarity!
bonyz, are you hoping to put together some pre-built modules to accomplish this task, or are you asking how to design a circuit that would accomplish it? Those are very different questions that deserve very different answers, and from what I read, I'm not sure which way you want to go. You have joined a forum where most of the questions generally have more to do with designing the circuit, and the answers you're getting from various people are going that way.
Assuming you want something like "an instrumentation amplifier in a box", that you can connect to and you're done, I think the answer is going to be "no." Instrumentation amplifiers are going to require a circuit around them to take care of all the various details.
If I understand what you're asking, and this is where things get a little vague, you have a common-mode voltage (that is, the non-signal part of the voltage) of something like 10mV, and then, on top of that, the voltage that you want to amplify a whole bunch with this microphone amplifier you have. I
think your concern is that you want to eliminate the 10mV or whatever it is, and leave only the tiny signal, and then feed that into the mic preamp. Here's the thing: the mic preamp can take care of the common mode. It's made to do that. It might not be really great at it, but it will be what it is, and I think it should at least be good enough to try out your idea.
With that, based only on my intuition about what you're doing, my suggestion is this: set aside the issue of CMRR for the time being; let the preamp do what it can about that. I think it's just getting you side-tracked, unnecessarily. Wire your signal between the "in+" and "in-" terminals on the input jack of the preamp -- the tip and the first ring of a TRSS 3.5 plug. Turn it on. See what comes out.
These other discussions about noise are related to maximizing the signal to noise ratio. Now we're into designing a circuit. The issue of "noise" is different, I think, than the way you've been using the term or thinking of it. We're talking about, let's say, "hiss", that's introduced by the amplifiers themselves. It has nothing to do with CMRR or the parts of the signal that you want to amplify versus those that you don't. It's noise, and we could call it hiss or something else, that gets combined with the amplifier's input signal, and then gets amplified by the amplifier's gain. Once it's there, mixed in, you can't get rid of it.
The bottom line of the whole thing is that the best (proper) way to arrange the gain through multiple stages is to have the first stage as high-quality, low noise, high gain as possible. Here's what happens: your signal gets amplified, perhaps by a lot, perhaps by 1000, and the noise mixed in at the input also gets amplified by 1000. For that reason, we hope, we wish, the amplifier's internal noise is very, very low -- which is to say, we want the first stage to be very, very high quality. And when the signal comes out, with that amplified noise mixed in, we have a big enough signal that we don't have to amplify it very much, if it all, in subsequent stages. So everything down the line could be unity gain, for example, and in those cases, we keep our "fat" input signal with relatively noise, and just add a very minor, unamplified bit of (internal amplifier) noise at each unity-gain stage down stream. That's the optimum.
The idea of "waiting until later" to take lots of gain means that the small signal and small input amplifier noise are mixed forever, and then in the high-gain stage, you add even more amplifier noise, and now you apply high gain to both of those bits of amplifier noise. It's not as good. So somebody might say "That's not how you do it. It's not the right way." Designing a circuit gives you control over all those things; putting boxes together gives much less control, but the principles still apply.
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