The first major part of "Secret of the Old Clock" is finding all four mirrors. You use them inside the carriage house, so you can go upstairs and start with the next major part of the game. The mirrors are...
1. Inside the den. You can get it at any time.
2. Inside Topham's house. You have to solve Topham's word puzzle, before you can get it.
3. Inside Jim Archer's area. You have to get the key from Waddell before you can get it.
4. Inside the carriage house. You have to solve the golf flag challenge, in order to get it, I think. Maybe the game lets you enter the right solution, without having beaten the golf flag challenge.
As I said when I started the speedrun routing, I don't know what the trigger is for "hidden passageway in the den can be found now". Hopefully, the trigger is early on in the game, so I can get the mirror in the den and discover the hidden passageway on the same trip. If not, I'll have to take two trips to the den, one for the mirror and another for the passageway. I think you have to use the hidden passageway twice regardless, but I'm not sure.