How can I open a stuck, overfilled kitchen drawer that doesn't have a horizontal entry point?
If there are drawers above or below, fully remove them and attempt to access that way, reaching around. 

If not, try pounding on the drawer face with a rubber mallet (or something else that won't damage the finish) to try to get the drawer contents to shift a little.

Finally, you can put a 90 degree bend in a chunk of wire (coathanger or whatever) and slip that wire (with the bent part horizontal) into the "tiny gap", then rotate the wire to manipulate the stuff in the drawer. You may want an opposing 90 dregree bend on the vertical part sticking out of the drawer to act as a handle. The bent part going inside needs to be shorter than the drawer is wide, of course.