wheatstone bridge drive by small current, issues?
You really don't have much choice here.  There are two options:


amplify
Sample using some technique such that the 6.6mV signal can be seen with enough precision.


Using a 3V reference, With 10 effective bits, you'll get about 3mV resolution, so your entire output range will be covered in 2 least significant bits.  That doesn't sound like enough! With a 24-bit A/D that give you about 17 effective bits, you'll get about 23uV, and your range will be covered in about 260 least significant bits.  That might be enough.  Does being able to divide your output range into 260 chunks solve your problem?

Now you have to consider your noise levels to make sure your SNR is OK.  You can use a smaller reference voltage, of course, but I haven't had much experience with reference voltages at tens of millivolts, which is where you'd likely need to be.

What is the sampling/measurement arrangement that drives your concern that 6.6mV isn't good enough??  I suggest you start at the other end.  What resolution, in terms of a pressure change, do you need to reliably identify?  Once you know that, you can design a measurement system that can meet that challenge.