Getting close to the end of the game now, I realized I forgot all about getting the cue for Bottom.
I tested it out. What you need to do is look at the picture, from Jane's podium. No big deal, you have to do that anyway, to get her to reappear after solving the pie puzzle. Once you've seen the picture, you can pull a lantern in the hidden passageway. That leads to Emily's room, and Emily isn't there anymore! You can then break into her room and read the note with the cue.
Obviously, I've got to put this into my speedrun route somewhere. Maybe right after finding the picture? That way, you go to Emily's room for sewing and note stealing.
Or I could completely reshuffle everything around, so that trip to the hidden passageway happens at the same time I do my initial trip to the hidden passageway. So that'd be...
1. Go to the upstairs carriage house. See clue for Marcel.
2. Get key from Marcel. Go to Jim Archer and use the key to trigger the sewing challenge.
3. Go to Lilac Inn. Learn you need a sewing needle.
4. Talk to Jane. Get assigned the pie puzzle.
5. Once you solve the pie puzzle, check the podium.
6. Enter hidden passageway for the first time. Learn it leads to Mr. Topham's, take money from the piggy bank, solve Creepy Corner's puzzle for a record AND spy in Emily's room to make her disappear.
7. Get needle from Jane.
8. Go upstairs. Sew, listen to the record and see the cue for Bottom.
Oh, wow, that would be combining a lot of things. That's super efficient, compared to my original plan of combining "find hidden passageway in den" with "find mirror in bird clock in den". Then again, you have to do all these things anyway, so maybe it only saves five seconds (or however long it takes to enter the hidden passageway and go back upstairs.) Either way, I'm making this my new route!
...Which means I have to go back and double-check everything along the new route. The joys of speedrunning!