From chapter 90, after the troll attack:

"I disagree. It's not thinking that gets people killed." The words were spoken in a level monotone, as though reciting lines from a book.

"Harry," she said, hardly even thinking as she said it, "there's nothing you could have done -"

Something flickered in Harry's expression. His eyes seemed to focus on her for the first time.

"Nothing I could have done? " Harry's voice rose on the last word. "Nothing I could have DONE? I've lost track of how many different ways I could've saved her! If I'd asked to have us all given communications mirrors! If I'd insisted on Hermione being taken out of Hogwarts and put in a school that isn't insane! If I'd snuck out immediately instead of trying to argue with normal people! If I'd remembered the Patronus earlier! If I'd thought through possible emergencies and trained myself to think about Patronuses earlier! Even at the very last minute it might not have been too late! I killed the troll and turned to her and she was still ALIVE and I just knelt next to her listening to her last words like an IDIOT instead of casting the Patronus again and calling Dumbledore to send Fawkes! Or if I'd just approached the whole problem from a different angle - if I'd looked for a student with a Time-Turner to send a message back in time before I found out about anything happening to her, instead of ending up with an outcome that can't be altered - I asked the Headmaster to go back and save Hermione and then fake everything, fake the dead body, edit everyone's memories, but Dumbledore said that he tried something like that once and it didn't work and he lost another friend instead. Or if I'd - if I'd only gone with - if, that night -"

--END QUOTE--

Just thought it was interesting how after hours of thinking and listing completely useless strategies like giving them communication mirrors (that would obviously also had gotten jammed), Harry still misses his most egregious sin by far- that even after remembering he can use his Patronus for messages, he still didn't immediately Patronus Dumbledore to Phoenix-teleport to where Hermione is and save her. Even Dumbledore instantly appearing after Hermione's death doesn't make him realize he should have called him earlier. Even when he realizes he should have called Dumbledore he doesn't realize he should have done it immediately, he just thinks about Dumbledore acting as a Phoenix tear dispenser after Harry already found and dispatched the troll.

(As an aside, pre-committing to getting the time-turner shell removed immediately after the troll attack then going back in time might also have worked as any regular pre-commitment to using the Time-Turner, but that's a more difficult to think of solution, like instantly remembering Patronus messages).

Is the hat sorting Harry in two houses just a coincidence? Or….