Decreasing ADC Reference vs. Amplifying signal
If you reduce the ADC Vref you will reduce the step voltage size of each bit you read.  A some point the bit voltage size is close to the noise level of the existing circuit.  You then need to add much filtering if you want a steady ADC reading.  You may not be able to reduce the noise in the existing circuit.

On the other hand if you create a new amplifier circuit and increase the level of the input signal you might be able to keep amplifier circuit noise low and keep the 1 bit voltage level well above the noise, (also using a higher Vref).

Note that the noise being compared here is the existing circuit noise, not the noise mixed in with the signal to be measured.  Even the Vref voltage will have some small noise in it.