If the face-down deck is ever emptied during the game, reshuffle all discarded bird cards to form a new deck.
This reshuffling occurs when a draw is attempted and the deck is empty, not immediately when the deck runs out.

This is supported by the following rule:

• "If the face-down deck is ever emptied during the game, reshuffle all discarded bird cards to form a new deck."
  (Page 9)

The context describes this in the section about drawing cards, indicating the reshuffle happens as needed when a card must be drawn and the deck is empty, not at the moment the deck is depleted.

Source: page 9

Source: Core Rulebook (pp. 11, 2, 4, 5, 9)