Disclaimer - For this discussion, I'm assuming zero wind, and that you cannot proceed forward and simply drive out of your position.
I think you have two options there. One is to reverse out, and the other is to turn and drive out. Both are going to require a good shove from the dock. Since you mentioned backing out, let's look at that first.
Have one of your more experience (and stronger) deck hands push the rear of the boat out. Ideally, your rear end is a good distance from the dock. If your rear will clear the boat behind it is possible to use a fast reverse (~1000 RPM) with a hard left turn. If your reverse speed is fast enough, there will be enough water deflected by the rudder to negate the crabbing effect of reverse. This is a game time decision.
An alternative would be a similar approach, but to inject a forward right turn after shoving off. In this case you would either have enough room for a second of forward right which would kick out the back end far enough to clear the boat to your rear. Followed by a more gentle reverse. You can also repeat this Forward Right, Reverse Left combination to slowly pivot yourself until you're clear of the boat to your rear.
There are also techniques of using your dock lines (spring line) and fenders if you desire to pivot your boat on the dock itself. This could be required if there is a strong crossing wind coming from the left. Attach/Hold a bow line to a cleat near the drivers seat, place fenders to protect your right bow. Use an idle Forward Right turn to drive against the spring line. This will rotate the stern to the left and clear of the rear boat. Then reverse out. There are some good graphics that can be found on this site ( http://www.cruising.sailingcourse.com/docking.htm
). Yes it's a sailing site, but a sailboat is a power boat around the dock. The animation is a little over half way down called Forward Spring Line.
The Forward exit uses the boats better left turning tendency to pivot the boat in place. This is a combination of Left Forward and idle Reverse Crab. Repeated, it should pivot you clear.