When first entering the Sand Zone, notice one of the tiles above you near the teleporter. Shoot through it, and find your new path. Note that this is absolutely necessary towards getting the fourth ending. If you walk up to Curly before shooting through there, then you won't be able to shoot through anymore, and the tile will be sealed off. The tile will be solid but shootable before you either shoot it or walk up to Curly.
After emerging from the two breakable tiles which previously appeared to have no significance, head into Curly's residence to find that your shortcut helped you beat her to her house. Enjoy the alternate cutscene, and then continue collecting puppies as usual. NOTE: It is important that you do the sleeping puppy last. Do all the other puppies first, or else this won't work. Collect the sleeping puppy after retrieving the other four, and go over to Jenka's house. But DO NOT give the puppy to her. Instead, walk out of her house after Balrog jumps into the ceiling, and make your way into the Sand Zone Storehouse. Now, if you got the alternate cutscene with Curly, and brought the sleeping puppy in, then something different will happen. Before Balrog manages to forcefeed the red flower to Toroko, the puppy interrupts him, and is further delayed by King as usual. But before he decides to charge at Misery, you show up. King and Toroko's fates are uncertain for some time beyond this point, but there is no boss fight with Toroko since she never ingests the red flower. So, why does this only work with the sleeping puppy? Because Nene's description says that her dreams have been known to "portend the future." Because of that, she knows something bad is going to happen, and contributes largely to the prevention of King and Toroko's deaths. Despite what happens, you still get tossed into the Labyrinth, and Balrog moments after, except without the blade.
Make your way to the clinic, and find Curly. You'll notice, however, that the dialog suggests that she will be able to recover without any medicine, and by simply resting. The physician still requests that you retrieve the Cure-All, but the alternate dialog does not imply that it is mandatory to help Curly get better. The reason she's able to get better faster is since you never fought her, the only thing she had to fight was Misery. When coming back into the Clinic, you will find Curly is gone, and has headed to the boulder chamber. DO NOT give the Cure-All to Dr. Gero. If you do, you won't be able to get the fourth ending. Instead, continue through the Labyrinth until Professor Booster falls down. Go down and talk to him, and you will be able to give him the Cure-All. Once you've reached this point, your path to the fourth ending is secured. If you want, you can stop reading and figure the rest of this out yourself, since it should be fairly self-explanatory what to do next, but you can keep reading if you want.
Once Professor Booster is healed, and he will repair the teleporter for you to get back to Arthur's house. But you will first have to notify Curly that there is a better way out. So you'll be able to go back up to the Boulder Chamber, talk to Curly, and get back to the Mimiga village. After getting back, do as Booster says, and check the seemingly empty Mimiga village, until you find Sandaime who is still in the Yamashita Farm. Talk to him, and watch the cutscene. You will then be able to go to the Plantation after the cutscenes have ended. Make your way up to the Jail, and find Sue, Jack, King, Mahin, and Toroko. Walk up to them, and King will tell you to get his blade. Go to the Sand Zone, into the Sand Zone Storehouse, get the blade, and bring it back to King so that he can break them out of Jail. Sue will then bring you to Momorin, and the Rocket parts fetch-quest will take place as usual with a minor addition at the beginning which was inspired by its partial pre-existence in the map's script.
You might notice that with these events, it is not possible to ride the rocket up to the Doctor's throne without getting the booster back, and returning the Mimiga mask. Before riding the rocket up to the throne room, it may be wise to get the spur and missile expansions from the newly dilapidated Egg Corridor now that you have the booster 2.0. So, when you're done taking care of any necessary business, ride the rocket up, and enter the Last Cave. When you get to the Balcony, you'll notice that those four frenzied Mimigas aren't there anymore. Hmm... wait... How many Mimigas was Curly raising? Four. And how many Frenzied Mimigas are usually there? Four. Hmmm... then where are they? When you're ready, head into the Throne room to face the final bosses. The dialog will be slightly altered, but nothing major. The only real big thing is that since you skipped the fight with the core earlier, and the waterway, the core is a new thing to you, and Misery explains it for the first time. Also, since you rescued Sue earlier, Misery doesn't have her, and she doesn't get possessed by the red crystal, so she isn't a part of the fight with the Undead Core. After defeating it, head out as usual, run through Sacred Grounds, confront Ballos, and enjoy the fourth ending.