I have sketched out a design for a bracket that dangles your phone upside down, in front of the screen and turns the phone front camera into a webcam that's much closer to where you are typically looking.
The interesting bit is that software on the phone displays the part of the monitor that's covered and creates a "seamless display experience" (slightly tongue in cheek).
Not perfect by any means but maybe worth creating a prototype of both hardware and software to see how it feels?