Rear doorbell not working
The transformer-button red wire  is supply hot or "R" in thermostat speak.
The white wire is "front button switched-hot".
The chime-transformer red wire is return/common/neutral or "C" in thermostat speak. 

You need to bring supply hot, and "rear button switched-hot" to the other doorbell.

...  presuming the doorbell has 2 different chimes.