Weirdly, talking to Cid didn’t progress the plot on its own! We had to aimlessly wander back to the middle of the school for Xu to show up and speak to us, instead. She told us to the observation deck, as a ship was approaching, and god knows who’s on board! Since this was an urgent situation, we went to play cards, and this time we did win the Fujin and Raijin card (from someone other than the person who first had it, as it happened)!
As for the approaching ship, the people on board appeared to be dressed like the people we had seen rescue the strange woman during the insurrection in the Garden, though they claimed to be SeeDs working for Sorceress Edea. Naturally this only confused and conflicted the party. These supposed SeeDs demanded to speak to the headmaster, and told him they had come for Ellone. Stranger still, Cid seemed to think that Squall would know Ellone somehow, and told him that she could be found somewhere in the school. We ultimately found her in the library, and as I had guessed, she was the mysterious woman who had shown up at the start of the game, etc (because who else would it be at this point?).