After the night of Halloween, Harry and Hermione became better friends. They mostly visited Hagrid together, which Malfoy refused to join because "the hut smells, and I don't want my expensive clothes to stink". She really could not understand why Harry was friends with such a snob.
Apart from that, she has been carrying out her own investigation into the door under the cerberus. On the night of Halloween, she noticed that Snape was limping, and his leg was bleeding. She connected the dots, and figured out that Snape was trying to get whatever the cerberus was guarding.
She read more into Greek mythology and tried to figure out what it is that Dumbledore's keeping from Snape. In mythology, Hades' Hound guards the underworld. Is there some sort of Underworld here in Hogwarts? That doesn't make sense.
She decides to share what she knows with Harry. In the hopes that he would help.
"Not a word of this to Draco." She makes him promise.
Harry theorizes that whatever is under the door, is whatever was in the Gringotts vault he visited with Hagrid. The one that got broken into a day after they went there.
"It was a top secret mission! Dumbledore trusts me enough to send me on such important missions! I trust Dumbledore with my life!" Harry got Hagrid talking. Hagrid is very sweet, but it's very easy to get him to talk. Hermione noticed.
"Yeah, and good thing we went there when we did! Did you hear the vault was broken into the day after?" Harry, cleverly, carried the conversation.
"Yeah! Good thing!" Hagrid agreed. "I've always said the only place more secure than Gringotts is Hogwarts! That's because of Dumbledore…" Hagrid said, confirming Harry's suspicions, who gave Hermione a knowing look.
"Unbelievable! Imagine breaking into Gringotts and risk getting caught just to get whatever was in the vault." Hermione tried, but she was too obvious. Hagrid looked at her, looked at Harry, and then smirked.
"I see what you two are trying to do…" He said. "Very clever…"
"Well, you've confirmed that the three-headed dog is guarding whatever was in that vault." Hagrid turned pale, and the smirk turned to an expression of horror.
"You know about Fluffy?" He asked faintly.
"Fluffy?!" Both Harry and Hermione were surprised by that name.
"Yeah! Fluffy is my dog. And I'm not telling you what he's guarding, because I don't know!" Hagrid said, crossing his arms and turning around.
"You don't know or you won't tell?" Harry teased.
"Does it matter? The important thing is that it's none of my business, and it's none of yours either. It's Dubmledore and Nicholas Flamel's business." Hagrid said, and then seconds later he realized he let slip another secret that he shouldn't have. "I shouldn't have said that…"
"Who's Nicholas Flamel?" asked Hermione, eager to find out.
"No no no!" Hagrid closed his eyes, covered his ears. "I'm not saying anything more."
For the rest of the visit, they weren't able to get more out of Hagrid, but what they got so far was good enough.