By now the hint system will be pointing to the next upgrade location and it's time to go get it.
Get back to the Gathering Hall if you're not there already. The first thing to do is scan the corners of the ceiling to find Shriekbats and eliminate as many as possible so you don't get any unpleasant surprises later. Blastcaps are also scattered around but these generally don't sneak up on you.
There are ledges on the left as you exit from the Save Station; scale these, going over the door to Gathering Hall Access, and keep going around to the right. When you can't go any farther to the right, turn around and scale more ledges that circle to the left. When you reach the corner there is a tunnel you need to use Morph Ball to get through, but first scan for Shriekbats again. When you come out of the tunnel you'll find that the bridge to the next door has collapsed so you need to jump across the gap. It's a bit far but fortunately there's a bit of the bridge still hanging on the right side, so aim for that and hop up. Take the door at the top of the steps.
This is a passage with Shriekbats similar to ones you've encountered before.
Energy Core Access
This is a passage guarded by Eyons. If you've gotten the Charge Beam then you can destroy them and get through in safety, otherwise stun them with the Power Beam.
This is a large and complex area but the toxic water on the floor prevents much exploration. There are Shriekbats in several locations here so watch your radar and scan for them as you go. Look for the door on the opposite side when you come in and scan it for a new Research entry. It's not the last chance you'll have for this one but they're uncommon and usually accompany bosses so you may have other things on your mind when you see them again.
Take the passage on the left, hopping over the toxic pools when needed. You'll pass a Stone Toad on the right. These are harmless unless you're in Morph Ball mode, and if you are they will swallow you and spit you out again. They are immune to the weapons you have at the moment. Go through the tunnel at the end of the passage.
Burn Dome Access
Follow the tunnel. You may notice a couple of drops and since you can't jump in Morph Ball mode that means you won't be able to go back for now.
When you enter the room a machine rises up from the floor in the center and a battle begins.
Before getting the Morph Ball Bomb the drone was guarding, get a scan of the locked door if you didn't get it in the Energy Core.
Expansions (Chozo Ruins)
Now that you have the Morph Ball Bombs you can use them to break through certain tunnel openings; these often scan as being made of Sandstone. You also have the ability to jump in Morph Ball mode. To do this, drop a bomb and remain on top of it until it explodes. The bomb won't harm you but the blast will send you up in the air.
You get a chance to try out the first new ability right away. Locate the blocked opening opposite the door where you cam in and drop a bomb next to it to blow it open. Follow the passage to find a Missile Expansion. Start heading back.
Burn Dome Access
There are two places you need to use the Morph Ball Bomb to jump into the next tunnel. Drop a bomb to jump, then steer into the hole before you start to fall.
By now the hint system may be pointing to a new location. This seems to be the only time it points to an expansion as opposed to an upgrade or a route to a new area.
You now have the ability to defeat the Stone Toad; let yourself be swallowed and drop a bomb while inside its stomach. Do this to open a hole into a part of the room you couldn't get to before.
There is a new device here; a Morph Ball sized indent in the wall at jumping height. First get a scan of it, then drop a bomb to reach it and be locked in, and finally drop a bomb inside to activate the device and drop out. In this case it powers up a pump to remove the water from the room.
The pump only works for a short time and you need to get to the next location before the water comes back, you should have plenty of time if you know where you're going though. Go through the hole to the left of the Morph Ball slot into the main part of the room. Go directly across and over a low wall to find another Morph Ball slot. Activate it as before; when you do the water will stay drained so you don't have to worry about it anymore. Some platforms in the middle of the room rise from the floor.
Again you have a time limit. Get back to the door where you came in the first time and the platforms will be in front of you leading to a doorway on the left. Follow the passage up to a third Morph Ball slot and activate it. This stops the timer and raises more platforms from the floor where they lock into place permanently.
Go back again to the door where you came in the first time, then hop up the platforms on the right to the door that was locked until now.
West Furnace Access
Just a tunnel to go through.
The majority of the room is not accessible until later, but you can go through a tunnel to the bottom part of a Morph Ball maze. Do a jump at the end to get out and enter the inside of the Furnace and pick up an Energy Tank, the go back the way you came.
Energy Core Access
You can bypass the Eyons here completely now. Notice the blocked tunnel under the main passage and use bombs to break through and pass under them.
Return to the Watery Hall as before and get to the ledge where you picked up the Charge Beam. If you didn't get it yet then you might as well do so now; see the previous page. There is a blocked tunnel here so drop a bomb to clear it. Then follow the tunnel. Use a missile to clear the door at the end and go through.
Follow the tunnel and use another missile to clear the door at the end. (It seems the game designers really wanted to make sure you had the missile launcher before you could get in here.)
This is a small room but there seems to be a passage going up. You can't do that now but there is a Missile Expansion behind a grate on the lower level. Use a missile to break through the grate and collect it.
Backtrack to the Main Plaza, the return to the Ruined Shrine where you picked up the Morph Ball. Get to the other side of the wall and fight off the Beetles that are still there. Find the blocked tunnel on the right side and use a bomb to clear it. Go in to find a Missile Expansion.
Go back to the Main Plaza again and go to the Ruined Nursery on the route you took to get the Missile Launcher. There is a maze of channels carved into the wall where you come in. The first starting point is next to the door; switch to Morph Ball and jump once to get into the channel, then roll a bit to the right and jump again to get go up a level. Drop a bomb here to remove a stone block. This doesn't allow you to go any further but it does clear the way above.
Drop down and scale the ledges to the second starting point for the maze. Roll in, then follow the maze to the right, jumping and clearing the stone blocks where needed. Eventually you reach the upper right corner where a Missile Expansion is waiting.
Continue on the the Ruined Gallery. Drop to the lower level and notice two holes on the right side of the room. These are the two ends of a tunnel; switch to Morph Ball, jump in one end and come out the other, collecting a Missile Expansion on the way.
Keep going, following the route you took to find the second Energy Tank. Stop when you get to the Vault since it's time to open it.
Go around to the far side and notice the three locking mechanisms; these look a bit like Morph Bomb slots and work more or less the same way except that two are covered with a grating and the third looks too high to reach. The first one is easy; just drop a bomb in front of it to clear the grating, then roll in and bomb again to unlock it.
The second lock is still pretty easy now that you have the idea. Jump to reach the grating and drop another bomb when you reach it. The bomb should clear the grating soon after you land again. Now jump in and set off another bomb to unlock it.
The third lock is a bit of a puzzle, and though the first two locks are meant to give you a hint, it's difficult figure out what you need to do if you're unfamiliar with the game. The idea is to do a jump from the top of another jump. To do that, you need to drop two bombs at ground level about a second apart. When the first bomb lifts you into the air, drop another bomb at the top. If you time it correctly, the second bomb will lift you back into the air as you land, and then the third bomb will lift you again to bring you twice as high as a normal jump. It takes a while to learn the particular dotted rhythm needed to carry this off, but mastering this double jump will earn rewards later in the game. Now that you can get into the third lock, drop a bomb inside to unlock it and open the vault. Grab the Missile Expansion as your reward.
You should now have a missile capacity of 55 which is more that twice what you had. But you should appreciate the difference in the upcoming boss battle. For now, save your progress near the Gathering Hall.