All credit goes to Electric Wizard's Steam Guide. Big spoilers obviously:
Go to the dog's store and choose to buy a random discount item, and sooner or later you'll end up with the Blue Gem. With this item in your inventory, explore the Seaside until you've see an event called LIGHTHOUSE RUMOR. Pay very close attention to the words "Blue Gem" in your inventory; they should be shaking very, very slightly. After this, explore the School and pay very close attention (if you're like me and have seen every event, you may be clicking through very quickly): you'll find an event called EMPTY CORRIDOR (OR IS IT?), and the words "Blue Gem" in your inventory will begin to vibrate again. Once this event is over, explore the Forest, noticing that the words "Blue Gem" are vibrating with every event you see, and eventually you'll get your secret ending