Now that we know that the future Layton is probably down by the river Thames, we need to go there and find him.
To the Thames[edit]
Return to the Bookshop and talk to Maya for Puzzle 076, which will net you the 
Helping Maya errand. Exit to the Chinatown Gate and talk to Horace for Puzzle 077. Move South and talk to Segal, who will recommend you to try out the food at the restaurant by the Thames, and remind you that searching for the head of the Family is a bad idea.
Move to the Riverside Hill and future Luke will mention that you can take the stairs down to the Thames. However, you can do a short detour back to Flatstone St. 2 now and talk to Inspector Chelmey for Puzzle 139. Return to the Riverside Hill and go down the stairs to reach a new location. Talk to the rabbit. It can talk, its name is Subject 3, and it won't let you pass until you solve Puzzle 078, which will net you the 
Carrot Patch track.
Once you solve the puzzle, Subject 3 will tell you he was abducted and became a test subject in a research facility. A particular experiment the facility often use him for is to put him in one room, do something that emits a loud noise, then cause Subject 3 to end up in a different room. Talk to Subject 3 again for Puzzle 079, who will mention that the facility is near a body of water.
The Thames Arms[edit]
Move forward, and talk to Beacon for Puzzle 080, which will net you a 
fork sticker and a Beacon's Hat errand. Beacon will also mention that the lighthouse he is guarding seems to be managed by machines. Go back and move right to the Thames Arms. Check the river repeatedly for Puzzle 145. Tap the door to enter the Thames Arms.
Talk to the Barkeep for Puzzle 081. He will tell you that a regular had just left. He will also tell us that that regular goes to the Thames Arms every day, wearing a different disguise each day. Since he just left, our target might be heading back to the Towering Pagoda! Move out and talk to Barton, who notes that Inspector Chelmey has been interviewing everyone in town to get any lead he can.
Move back up to the Riverside Hill to encounter Belle. She will tell you that the Family seems to be searching around for us. If the Family realized that Layton and Luke have come to stop them, you'd better go to the Towering Pagoda before they ramp up their security!