What quantity changes when the signal is amplified without changing SNR in this case?
Actually the system's S/N will likely change if you amplify the signal (but stay within the linear range) to an ADC.  That's because the quantization noise of the ADC in relation to the signal will decrease.  The closer the signal's peak is to 10V, the more "steps" that are available to be used to represent the signal's levels between min and the new higher max.  The more steps available, the less the proportionate rounding or truncation errors, statistically.

The case where this would not be true is when the signal is stepped and all its levels match the current ADC quantization.