Than why 9-bits?
Why 24 bits?
Typical "Absolute accuracy (Including INL, DNL, quantization error, gain and offset error)" of the AtMega328 is 2 LSB. One more argument for using external ADC.
2LSB absolute accuracy in your opinion means 10bit ADC degraded to 8bit or what?
Look, OP did not mention *any* accuracy specs. Could be so that 8-bit ADC range is completely fine for what he is planning to measure.
In my opinion, considering time / costs/ reliability - external ADC is the only option.
The question is what is better: using external ADC or build an amplifier. Having 10 bits internal ADC and spending a lot of efforts for re-scaling/ offsetting to get what, just 1-bit (!) more?
O'k, you may buy MCP602 (offset 3.0-4.5 mV ) for 0.5$, and get may be 9 bits (see offset). OR spend 2$ and get 16-24 bits (the more the better, isn't it). The choice is obvious.
For pure academical purpose, putting economic aside, could be good exercise, atmel has AVR465 app note on topic.