Near the exit of the pyramid, there is a man with a spear. Speak to him to get the key to Dungeon Man. You will need this soon, so don't forget it.
Just north of the guy who gave you the key is a man with a camel standing next to an oasis. He will sell you some really powerful stuff, but it's expensive, so you might not be able to get all of the things that you might want. The Multi-Bottle Rockets are the strongest type of bottle rocket you can get for Jeff.
From the oasis, head northwest. There are some tough monsters here, and you're also at risk of heat stroke, so don't worry about exploring. There's not much to find here anyway.
After going northwest from the oasis, you'll eventually reach Dungeon Man. He's a tower with arms, legs, and a human-like face, and you can talk to him. It turns out that he's Brick Road, the dungeon maker whom Jeff met on his journey through Winters. Through Dr. Andonuts's scientific wizardry, Brick Road has been made a permanent part of this tower. Using the key to the tower, you can go in.
The music here is super-weird. At any rate, Dungeon Man is a quirky maze, much like the Brick Road maze that Jeff went through, except that it's several times larger. There are strange gifts around, and relatively weak monsters. There are also some benches that you can sleep on to heal up. There is even a doctor's office in case you get an ailment that you can't heal, like mushroomization. In addition, there is an ATM.
The closest healing bench is to the left from the Dungeon Man entrance. Once you're ready, go right. When the path splits into three, you should go to the right first to get the Molokheiya Soup.
Return to the three-way split and take the middle path. At the bench, turn left. You will reach a gift box with a Cup of Lifenoodles.
Finally, return to the three-way split and go left. After the sign, take the path that leads downward. You will reach a gift box with five dollars inside. From there, go up the ramp to the higher level, and go right when the path splits to get a Snake. Go back to the split and go up. There is a Super Plush Bear at the end.
Return to the place where you got the five dollars. Go left and you will reach four ropes. Go up the third from the left.
You will reach a sign, and there will be another sign a short way below that. At that second sign, go left along the top wall to get ten dollars. There isn't anything else interesting on this top ridge except for a Good Night Bench if you need it. But otherwise, go back to those two signs that are close to each other, and go south from the rightmost sign.
Hug the wall until the path splits such that you can go up or left. Go up, then climb the short rope that you reach. Go south from it to get a pizza, then go north to get a Sudden Guts Pill near a sign. Go farther north, beyond a rope, to get a PSI Caramel.
Then, go back and climb that rope. There are no enemies up on this floor (not any that will fight you, anyway). Go right from the rope and you'll see some Gruff Goats on display. Go south from there and you'll see Slimy Little Piles on display. Go left from the Piles, staying along the bottom wall, to get a Wet Towel near a Good Night Bench.
Go back to the sign for the Slimy Little Piles, then get onto the ledge that is to the left of that sign. You'll reach some ducks on display, and there's a rope nearby. Climb it.
Go north from the rope to find Brick Road himself. He is a permanent part of Dungeon Man. He'll heal you up and then say that he'll agree to follow you. Now you just need to get back outside.
Go south, and drop down into the first hole that you find. Keep falling down the holes that you encounter, being sure to grab the IQ capsule that you see along the way. Eventually you'll end up close to the exit of Dungeon Man. Go outside.
Walk around and Dungeon Man will follow you. In fact, he'll join in whenever you fight any enemies. He's extremely strong, so you might want to go around fighting enemies with him to gain experience more easily.
When you're ready to progress to the next part of the game, walk south with Dungeon Man and go through some palm trees in a narrow space of land. Dungeon Man will get stuck. Go east and talk to the spear guy to learn that you need a submarine in order to get to the Deep Darkness from here.
Talk to Dungeon Man and go back inside, then make your way back to Brick Road's face. He'll tell you that he has a submarine in his collection. To reach it, go south, past the hole that you dropped into before, and around to a second hole that was previously blocked by the sign.
Eventually you reach the first floor. Go south, then east, to get a Talisman Ribbon. From there, go left. You'll find Dungeon Man's collection. Be sure to use the Instant Revitalizing Machine here to heal up. Then check on the submarine. Jeff will see that it still works. You'll automatically get in and cruise on over to Deep Darkness.