I am assuming you have already beaten the game. Mild spoiler warning.
If I have made any mistakes, please say so in the comments. ( I think I might have messed up with the key location for the School.)

Enter the docks doorway from the nexus. Use the umbrella to jump left past the fisherman, and glide as far left as you can.

Go to the mannequin room. Go right past the jumping segment and gate, and into the room with all the boxes. Above the door you came in through, there's a secret passageway. Through there, there is a dark room filled with mannequins. Equip the lamp, and go left until you pick up a Rooftop Key. Go back to the entrance of the mall, and go in the door next to the entry point. Go up the elevator, and use the Rooftop Key on the door to the right. Go through that door, and go right.

Enter the door from the Nexus, and go left.

After trapping the angry orb in the Ghost Town, one of the doors in the middle alleyway near the entrance will open. Go in there.

One of the rooms in the destroyed school will have a chalkboard with a hand symbol on it. Use the hand effect near it and pick up the purple item (a key). Go back to the regular school, and go to the top floor. Go in the previously locked bathroom and touch the jellyfish.

For this, you will need the two new effects: the Hat and Scarf and the Mask of Fear.
Head to the sewers. First things first: equip the Hat and Scarf, and use the Umbrella. This will turn the slowing water into a platform. Then, walk down to the lower middle section. There’s a door you can enter. Inside is a room full of the red laser eye guys: wearing the Mask of Fear will kill them. Walk all the way through and touch the jellyfish!