First, you solve the puzzles:
| An Area With Big Stones and Rocks | MONTERA |
| A Pile of Blocks | PARTY |
| Audio/Visual Equipment Closet | CHESTNUT |
| Denver Botanical Garden | LEMON |
| Hall of Justice | FOREST |
| In the Park | BLUE |
| On The Runway | COCKTAIL |
| The Islands of the Cartesian Sea | PIRATE |
| The Sixth Floor Games Lounge | MAGENTA |
Then, each of the puzzle answers has an answer which is either a color or a hat. Gathering the answers by type, you'll notice that the answer lengths are all distinct, which gives an ordering for each type:
| A Pile of Blocks | PARTY (hat, length 5) |
| The Islands of the Cartesian Sea | PIRATE (hat, length 6) |
| An Area With Big Stones and Rocks | MONTERA (hat, length 7) |
| On The Runway | COCKTAIL (hat, length 8) |
| In the Park | BLUE (color, length 4) |
| Denver Botanical Garden | LEMON (color, length 5) |
| Hall of Justice | FOREST (color, length 6) |
| The Sixth Floor Games Lounge | MAGENTA (color, length 7) |
| Audio/Visual Equipment Closet | CHESTNUT (color, length 8) |
Finally, indexing into the answers (first letter from the first word, second letter from the second word, and so forth) in each set, you get PINK (from the hats) and then BERET (from the colors), and PINK BERET is indeed the present for Piglet.