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The application note in the post above will tell you all about the drawbacks of rail to rail amplifiers. The good point is that with low supply voltages you will still get a good input common mode range and output swing. With ordinary amplifiers they output can only go to withing about 1.5 V of either supply. With a 3 V supply there will be no output swing for instance.
Its benefit is to be able to work with full range of power supply.For example if somethnig is not rail-to-rail may mean for 5V system that you can limited with Minumum working voltage: 0.2V and 4.7V maximum voltage.(Say you can not meausure voltages below 0.2V if this will be an ADC)Note that these values are just examples...
I think that you can use a charge pump to improve your desing. Using this technique you can provides the local supply voltage for the input differential pair, so the input signal can swing between the rails (rail to rail).
I have a paper that explain this technique and use it in a design of a CMOS rail-to-rail input operational amplifier with THD performance of -90 dB which is suited for high-quality audio systems.