Will noise performance improve if I reduce the filter bandwidth?
[edited by: EstiS at 7:39 AM (GMT -4) on 20 May 2021]
Noise may reduce at low gain settings too. The amount of improvement will depend on the signal chain and filter implementation.
A simple example of this would be a unity-gain buffer with too much bandwidth. This is low gain, but filtering could still improve the noise, perhaps quite significantly! One may argue that a different amplifier should be used in that case. That may be true but there are possibly other design factors constraining the choice of amplifier.