no, because if you play at lower volume, that simply means that the quietest parts are closer to (or underneath) the threshold of hearing.

I see it this way: let's say 16 bits is needed to represent the entire discernable dynamic range between the threshold of hearing and the threshold of pain. if you turn the volume to 50%, then you throw out 8 bits, but you also only need to represent 8 bits worth of hearing range.

Why woulda few extra bit in the source data help there? A DAC with more bits would, but that can easily be scaled from the existing data.

An aside: I've never understood why the logic around digital signal-path volume adjustment isn't "keep the volume number around all the way to the end; throw the signal through the DAC at 100% gain, and then attenuate the signal on the analogue path using the digital volume setting." Uses more power, I suppose.