Ah... So the noise created also depends on the fretted notes being played? That would suggest something much harder to diagnose and correct... Harmonic induced sympathetic resonant frequencies. By touching behind the saddle that might just be enough damping to mute what is in essence a design or build flaw.
If it is that it's not easy to correct. I had a problem on my Martin acoustic that was and is annoying, although I did address it partly. I played specific notes on the G string or E string and it set up a resonant frequency on the A string. Likewise touching the A string at saddle would dampen the resonant coupling.
This article explains some of the factors:
http://www.phys.unsw.edu.au/music/guitaracoustics/construction.html
String choice can also affect it but less so than the construction. That said changing string gauges or from phosphor bronze to bronze flat-wound is nice and quick as an experiment.
A good luthier might be able to source it and charge you lots or you can experiment with damping in the sound cavity (other approaches are more permanent and more expensive. Whatever you do (damping, strings, bridge mounting etc. will affect tone. The trick is to do something that pleasantly alters it, and for the layman that takes trial and error.
Chances are it will be some damping material (experiment with type), placed somewhere on the backboard or soundboard towards the tail of the guitar. Changes to the saddle mount are more likely to change the volume and tone of the higher notes, the E1and B strings. The annoying part is the experiment usually involves loosening or even removing strings for each experiment. Persevere and you should be able to improve the issue without destroying the overall tone or volume.
