After you break into Hilihili, you have a straightforward segment. You take the leaf to Camp Quigley. Go to the table. Test the frass and the leaf to see they don't match. Specifically, You click the frass, then the eyedropper, then the tube, then the leaf, then the eyedropper, then the tube.
Go to Quigley and talk to her. Grab the container in her pack, then go back to Camp Quigley. Use the container on the table to test it. You want to test the frit. flag beetles, which is the third-from-bottom option. It looks like a peeled banana.
You return to Quigley. You can choose one of two conversation options here. My timing showed the first option is .07 seconds faster, so yeah, I'm gonna say it doesn't really matter. Quigley gives you the nose ring, so you can go to Kapu Cave and start the endgame challenges.
I noticed that the World Record speedrun had Nancy visit Kapu Cave earlier in the run. That is, Nancy went to the part of the cave with webs in it. This triggers a conversation with Quigley about the webs. Since I don't go to the end of Kapu Cave until this point in the game, I skip that optional conversation. Minor timesaver!
The first puzzle at Kapu Cave is moving the teeth. The starting positions are always the same. Click the pieces like this. The number is the number of times you click.
1 1 3 _
1 2 _ 1
So...you save time by not clicking on either of the two teeth that are already in place. That's basically the only timesaver in this segment. Brilliant.