This prison is a little bit trickier as it is large, lot's of outdoors, and a lot of inmates. The good news is that there should be a plethora of items when you search through desks.
Stats Needed:
Intellect 60
Items Needed:
2x Talcum Powder
2x Tube of Toothpaste
2x Lighter
2x Comb or Toothbrush
4x File
3x Roll of Duct Tape
1x Guard Uniform
1x Rock Hammer
This escape requires chugging through 8 consecutive favors from the same inmate until they give you a special item. At the first roll call, start accepting favors until you get one called "Rock-Hammer Hard Place" that is asking you to find a book. Make sure you know who it is that gives you this favor to make the subsequent steps easier. This inmate will continually ask you to do stuff from finding items, beating people up, disposing of stuff, and making weapons. Use the favor icon on your mini-map to find all the items to get through these favors as quick as possible. When you get to a point where to need to make a weapon for the favor, you will have in increase your intellect. Put away all your contraband items since there is a metal detector on the way to the library. Head to the south-central building to get your intellect up. Once you have completed all the favors for the inmate, they will give you the Rock Hammer which is contraband, so make sure to hide it in your desk.
Now you can focus on obtaining all the other common items by rummaging desks. Once you have the Talcum Powders and Tubes of Toothpaste, combine them to get 2x Wads of Putty. Now you can start beating up guards. I wouldn't go out of your way to beat up guards, as you will likely get in trouble while rummaging desks. You are looking for a Purple Key and Yellow Key as well as a Guard Uniform. When you find the keys, make a Key Mold by combining the key with the Wad of Putty, then replace the key on the guard's body. Be careful with this section when you're beating up guards. If you're in the opposite building from your cell, you won't be able to stash your contraband items because you will get shot if you go outside. I recommend hiding the key molds in particular in a desk, get beat up to lose your heat, then go retrieve them. Once you have both Key Molds as well as the Lighters and Combs/Toothbrushs, you can make the keys. Combine the Lighters and Combs/Toothrbushs to get 2x Molten Plastic. Combine the Molten Plastic with both Key Molds to get a Plastic Purple Key and Plastic Yellow Key. Hide these in your desk.
Now it's time to make your escape tool. Find the Files and Rolls of Duct Tape so you have enough to make the best cutters. If you need space, you can make the cutters first, and upgrade them as you find the additional items needed. Combine 2x Files and Duct Tape to make Flimsy Cutters. Combing this with an additional File and Duct Tape will get you Lightweight Cutters. Again, adding a File and Duct Tape will get Sturdy Cutters. Now you are ready to escape.
Grab your Guard Uniform, Both Keys, Sturdy Cutters, and Rock Hammer. Wait until the next lights out, and put on your Guard Uniform. Head out of your cell and towards to southern end of the cell block. Out the purple doors, there is a giant statue. Just to the east of this is a sewer cover. Use the Rock Hammer on this sewer cover when there are no guards around to open it up. It takes two hits to do so. Then, drop down and head north until you reach a fence. Cut through with your Sturdy Cutters in four hits and continue. When you can, head west through a long tunnel until you hit another fence. Cut through this, and at the end of the tunnel is a ladder up to freedom.