Secure storage
You're now in the storage room where Ashley was being kept prisoner. Grab the Shotgun Shells hear and start for the exit. Something flies in through the window, pick it up to get ”Paper Airplane”; it's from Ada and it's a clue to how to get out of the complex.
More soldiers are waiting outside the cell so leave Ashley inside and defeat them. A cheap way to do it is to shoot at them through the window in the door.
Head south back to the waste disposal area as Ada hinted.
Waste disposal
You immediately have to fight soldiers guarding the security panel; have Ashley stay in the entrance, then stun the soldiers that are approaching then snipe the archer and the battle is practically won. At some point a soldier will appear and shut the door to the waste disposal room, so now you need to open it again.
When you open the door more soldiers are going to come in through it, they're going to do it very quickly and they're going to come straight for you. So some prior planning is called for. First you need to find a safe place for Ashley, away from the action but far enough away from a door that she can't be carried off quickly. The best place is probably in the corner with the computers. (Maybe she can play some video games while she's waiting.)
Position yourself at the security panel so you're facing the door as much as possible, then activate the switch and before you know it the soldiers will be right in your face. A useful trick here is to interrupt the short cutscene showing the mercenaries entering the room; when you do this the mercenaries will still be entering the room rather than already surrounding you when you get control again. An Incendiary or Hand Grenade is dangerous to use when enemies are very close, but they may work if you can get the timing right. An alternative is to use a Flash Grenade to stun them, then get over to where Ashley is waiting and make a stand there. Yet another alternative is to just fight them where you are; in this case the TMP has the required combination of speed and firepower.
When the battle is over, go through the door you just opened head down the hall into the waste disposal room. There is a cutscene where you trust Ada’s message and jump into the garbage chute.
Trash heap
You land safely, but Ashley spots a regenerator lying dormant next to her, though only you know what it is. Get the lever just ahead to raise the gate, but the monster will wake up and you probably won’t be able to kill it before it gets too close for safety. So just turn and go into the next room to gain some time; toss a Flash Grenade to slow it down if you have a spare. There is a bin for Ashley to hide in on the right, through she should be fairly safe if she stays behind you. Defeat the monster as before, but a trick here is to go into the next room and pull the lever next to the doorway you just went through. This closes the gate behind you so the monster is trapped behind it and you have some extra time to snipe it, though it will eventually open the door again. Note, there is an exploding barrel here but these don’t seem to affect regenerators much.
Pick up the monster's 20000 PTAS drop which should be very useful, then get Rifle Ammo on the floor. Go into the next room and call Ashley. With her help you can push a large bin into the water and creat a bridge. You might want to have Ashley hide again while you defeat another regenerator which comes down a passage on the left. You can use the closing gate trick here as well, though in the other direction.
Get Rifle Ammo and a Green Herb on the other side of the water. Then cross a bridge to where a large bin is blocking a doorway. Go through into another room and then through the door on the left.
Foundry
You enter a passage filled with trash and debris. Soldiers attack right away including one with armor. Get two barrels on the right though there is a good chance that they will have already been destroyed in the battle. Grab a Green Herb just beyond this.
You come to a ledge over a room with a pool of lava, but scout it out before dropping down since there will be a battle here. The room on the far side looks tempting as a defensive position, but it has two doors making it hard to defend. A nook just to the left of it makes a better choice. You might think that Ashley would be safer if you left her on the ledge, but soldiers follow you in that way so it won't work.
So, keeping Ashley with you, drop down into the room. Ashley points to an opening on the left but that's not very useful at them moment since soldiers appear. Start by tossing an Incendiary Grenade at the first group of soldiers coming in from the left, then circle around to the nook to make your stand. There will be armored soldiers and archers among the enemies. Remember to aim for the head with the armored ones, or for the feet to slow them down. Stun the enemies nearby and snipe the archers when you get a chance.
Use the TMP or a shotgun as your weapon of choice. A shotgun is nice here since you can blast enemies into the lava. Toss a grenade if things get overwhelming and don't forget to keep your health up.
When the music stops, signaling that the battle is over, get the Handgun Ammo and Red Herb from the control room. Then have Ashley operate the control here to swing a wrecking ball and break open a hole in the wall. You can do this yourself but having Ashley do it allows you to prepare for another wave of mercenaries that drops in from above in when the way is open, stay close to the control panel since it will give Ashley a bit of cover during the battle. This battle isn’t as bad as the previous one. Afterwards, go through the door behind the hole you just opened up.