I don't understand where it is hitting, you mean down in the bottom of the hole when it opens the leading edge of the door is hitting in the bottom of the recessed area of the quarter? If that is the case you need to move the pivot point nearer the edge so it swings out more like a gate, if the hinge pivot point is back away from the edge the front swings in like a door the car swing in the front edge. So moving the pivot point forward is what it sounds like you need.
And that pivot point needs to be as close to the skin of the lid as possible. The further away from the skin the more the lid will move forward before going in at the front needing a wider gap.
Is that where you are, am I understanding this?