Find a rail-to-rail input opamp these can accept input voltages very close if not equal to supply rails.
You could also use a low cost mainstream opamp like 741 and bias the whole measurement circuit off of ground. but extra components necessary will likely offset the extra cost of a r2r or inamp anyhow.. but its another way. Look at LM3 schematic, you might be able to bias it off of ground so its lowest output is within range of your amp..