A common volume pot that we use 99% of the time is just an attenuator so you'll get unity in the full clockwise position making it pretty pointless. You'd need to add some sort of boost daughterboard to do what you suggest with the volume pot varying the gain from the stage and so controlling the output volume. Something simple like this would do the trick:
The opamp is a single like an LM741 or TL071 and if you use a 10K volume (gain) pot that will give you between 0 and 4x amplification. If you don't want that much to keep unity around the central position of the pot, swap the 3K3 for a 4K7 which will give you up to 3x, or a 10K will give you a maximum of 2x