I'm a bit confused. I have no idea what a dollar bill weighs but I imagine it is very much less than a coin so I would expect you would have to change the physical range of the scale (a stronger spring or shorter lever) to weigh coins.
Assuming you are using the default 5v reference for the Arduino IDC 422 equates to just over 2 volts and 900 equates to just over 4.3 volts. That seems to be the range produced by your scale and, as you say, it is wasting some of the resolution of the ADC. But that is a different matter from the range of weights the scale can measure.
If you discovered that it runs past 1023 before the scale is fully depressed that would suggest that the upper end of the scale output exceeds 5v and a voltage divider (or adjustment of the amplifier gain) could be used to match the output to the ADC. But clearly the scale does not produce an excessive output.
Without having the specifications for the load cell I can't think of anything else to say.
...R