Bathroom sink Overflow
If you can get a plunger to seal over the overflow hole, you might be able to force some air in there, or vacuum some air out, and dislodge the clog.  

These things are very inexpensive and might work where the wire hanger did not.  Some are more flexible, some are pretty stiff.  You might also try a flat steel drain snake, just be careful and don't apply too much force.