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This puzzle is now solved. Highlight the white text inside the block for hints ranging from vague to specific...

Hint 1:

From the previous puzzle, For the Want of a Clue, we know that we need to KEEP A WATCHFUL EYE OUT FOR A CLUE. These are hidden as "clocks" in 12 of the images accompanying the puzzles previously encountered.

Hint 2:

There are 3 clocks per suit, each showing an hour hand. These times are transformed and placed in the Bully Pulpit grid, from left to right, in the order they were encountered in the story. The first is in 8 Diamonds.

Hint 3:

The cards you are looking for are 8D, 4D, AD, AC, 9C, 10C, 7H, 5H, 2H, 4S, 9S, JS.

Hint 4:

The times you are looking for are as follows:
8D - 7:00 - Target & Arrow Shadow
4D - 5:00 - Whetstone & Spoke
AD - 3:00 - Window & Triangular Shard
AC - 6:00 - Ring & Lantern Holder
9C - 5:00 - Chessboard & Hourglass
10C - 8:00 - Sphinx Headdress & Glowing Gem
7H - 1:00 - Colleen's Eye & Light Glint
5H - 2:00 - Pocket Watch & Hour Hand
2H - 7:00 - Ring of Shell Tubes & Water Pouring Out
4S - 11:00 - Obol & Notch
9S - 5:00 - Star Globe & White Star
JS - 6:00 - Circle of Colleen's Skirt & Dagger

Hint 5:

The boxes in the Hidden Goal parchment are raised for AM times and lowered for PM times. How does that affect the transformation?

Hint 6:

PM times occur 12 hours after AM times, so add 12 for the lowered boxes. This gives you the following times:
19 - 5 - 3 - 18 - 5 - 20 - 1 - 14 - 19 - 23 - 5 - 18

Partial Solution:

Transliterating these 24-hour times into letters, you find that "The Hidden Goal" is a:

SECRET ANSWER

But that is not "Where to Exit the Maze"... one more step is required for that. This is where the second part of the couplet extracted from the previous puzzle (For Want of a Clue), AND AS MY PEOPLE EXIT SO CAN YOU, becomes important.

Ultimate Solution:

A little Greek historical context: Asterion, the Minotaur of Greek mythology, dwelt at the center of a labyrinth built by Daedalus and Icarus for King Minos on Crete. In fact, the word Minotaur is derived from Minos + Taurus (bull).

So, how does this help? Since the people of the Labyrinth were from Crete, what happens when "MY PEOPLE EXIT" from the hidden goal? The people of Crete are known as Cretans, and when the Cretans exit (are removed), we finally find where to exit the Maze...

SE(CRETANS)WER

So the Quaices ran through the grate below that mechanical waterfall, and raced back up to climb out of the SEWER in the alley behind the Upper Wolverhampton library. SEWER is the final solution for where to exit the Maze of Games.