> There's a #%&! finger covering up the screen now!

At which point I use the abductor pollicis brevis, adductor pollicis, etc. to do something quite amazing: I move my thumb out of the way. Alternatively, I can use the same muscles to scroll the screen to a point such that I can see any obstructed content.

The benefit of this method is that I maintain my original viewing angle. I also have finer motor skills in my thumb than in my wrist, and thus I suspect I have greater control.

(FYI: Might want to tone it down a bit. You sound like an astroturfer.)