Brainstorming about Snape, Orpheus, and Eurydice
Delphi's next detention with Snape needs to involve a conversation that will ultimately lead to Delphi letting it slip that she knows about Fluffy living in the forbidden corridor. His reaction, then, should prompt Delphi to ask Hagrid about the inexplicable flute he gave her for Christmas (which he should reveal was given per Dumbledore's suggestion).
One possible starting point for the conversation should be Snape asking Delphi once again to confess what she was doing out of her dormitory after curfew. Perhaps this time Snape phrases it in such a way that Delphi realizes he genuinely doesn't know that she's not been in her dormitory for months now, and Delphi gets a hint that it wasn't Snape who tossed her room after all (it must be assumed that teachers are not permitted in the children's actual bedrooms, as they are never once seen there in seven books) (perhaps this is when it clicks for Delphi that it wasn't Snape who did that but the prefect, who Snape tasked with keeping an eye on Delphi and who has been successfully pretending that Delphi's been a well behaved student still living in the dormitory).
In any case, Delphi definitely isn't willing to admit that she's been squatting in an abandoned classroom, and she refuses to tell him what she was doing. He pressures her, getting rather viciously mocking, and when he insinuates that she could get herself into trouble by wandering the school, she quite unthinkingly quips, "What, like nearly getting my leg bitten off by a three-headed—?" before realizing to her horror what she's just let slip.
Snape seizes upon this mistake. He demands to know what she meant by that, and Delphi can't think of a good lie to compensate. He tells her that if she is caught anywhere near the forbidden corridor, he will personally make sure that she is expelled; further, he warns her that if any other teacher so much as implies that she should go near it, she must report it to him immediately.
This confuses her. Snape is the one who wants to get past the dog, isn't he? What other teacher would instruct her to go near a cerberus? Does he think someone would task her with retrieving the Philosopher's Stone? Mind racing, Delphi makes her promises to Snape, and then goes off to investigate.
She stops in the library and grabs literally everything she can find that so much as mentions the Cerberus. Among these books is a book of legends, which includes a retelling of the Orpheus and Eurydice myth from a wizarding perspective. In the story, Eurydice is killed by a snake on her wedding day, and her husband Orpheus travels to the underworld to rescue her. To reach her, he must past the cerberus, which he charms with the music of his flute. But Orpheus is warned not to look at his bride before they reach the surface, or else she will be lost to him; he can't resist checking on her, just one little peek behind him, and so she disappears forever.
This makes the connection for Delphi: music is the cerberus's weakness. And suddenly Hagrid's Christmas gift makes sense.
(On a more meta level, this also foreshadows what will happen in the "underworld". Harry follows Voldemort down into the final chamber, and Delphi can't follow him until Dumbledore removes Snape's fire spell. But Dumbledore warns her to trust him and to stay away from Harry and Quirrell; when she doesn't listen—when she metaphorically looks back—she gets possessed by her father's severed spirit and nearly loses herself—and Harry—forever.)
(Canonically, of course, Through the Trapdoor and The Man with Two Faces have shades of being a very loose retelling of Orpheus and Eurydice with Harry as Orpheus and the Philosopher's Stone as Eurydice. Harry gets the Stone in his pocket, meaning that he can take it to safety, but if he had taken it out of his pocket—Orpheus's glance back at Eurydice—it would've given Voldemort the opportunity to take it from him without touching him.)
With her newfound suspicion about the flute—and the implication that there may be some other player in this game, considering that Snape apparently suspects someone is manipulating her into going after the Stone—Delphi goes to interrogate Hagrid about his gift to her (and to Harry) and may or may not come away from the conversation with the knowledge that Dumbledore is the one who's setting everything up. If nothing else, she quite likely suspects.