Op Amp: signal measurement that is below its common mode input range
I'm looking for an instrumentation amplifier and most of them I see
  have very narrow Vcm range with respect to their supply.


Consider using the AMP04 instrumentation amplifier: -


  Although not tested and guaranteed, the AMP04 inputs are biased in a way that they can amplify signals linearly with commonmode voltage as low as –0.25 volts below ground. This holds true over the industrial temperature range from –40°C to +85°C.


It operates from a single 5 volt rail too and, the output can swing down to a couple of mV. However, a limitation is that from a 5 volt supply, the upper input commonmode range is limited to +3 volts.

There may be better alternatives - I suggested this one because I use it in a design.