Expect to encounter resistance from Slate's men after getting back to the main hall. Eliminate all enemy units and then proceed towards the final exhibit.
Continue exploring the initial corridors of the exhibit and you'll soon come across a locked door. Just as with the previous story-related doors you won't need any lockpicks here, because Elizabeth will use her hairpin to open the door.
There's a room with a fireplace to the right of the door. Examine a chair located under Lady Comstock's painting to find Voxophone #31. Pick up this collectible and resume your journey, proceeding to the next rooms.
Once you've arrived at the courtyard you'll find a locked gate and only Elizabeth will be able to squeeze through a small opening. Thankfully Elizabeth's supernatural abilities will begin to manifest more often from now on and she'll be able to open tears for you. Start off by opening the nearest tear in order to summon scaffoldings you will be able to use to get to the other side of the gate.
The courtyard is guarded by a squad of Slate's men and a rocket turret. Notice new tears around this courtyard and you'll be able to use them to summon hooks or a barricade with an allied turret. Remember that opening one tear automatically closes the other one. I recommend that you practice to effectively use tears, because they will become an inseparable part of the tactical aspect of the game. Tears will be available in most of the battles you'll take part later in the game and they'll allow you to achieve victory in different ways depending on your invention or preferred style of play.
After leaving the courtyard you'll get to the gift shop's vestibule. There are vending machines here and you'll also uncover Vantage Points #19 and #20. Watch all the materials and then enter the store.
There's a Vantage Point #21 in the middle of this room. Watch a new material and prepare for battle.
This is the moment when a grenade launcher mentioned earlier in the walkthrough may come in handy, because you're going to be attacked by a mechanized patriot armed with a crank gun. The weakest points of the patriots are their backs and you should be aiming at flywheels marked in red.
Once you've defeated the patriot search the shop and ask Elizabeth to unlock a door.