Here is the official answer which I posted in another thread too.
The "must find and defeat" condition begins when the "Don't Leave the Path" card takes effect. If the Spawn is in play at that time, the "find" aspect is immediately satisfied, they simply have to defeat it. A previous defeat, though, will not satisfy the "defeat" aspect of the card.
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Simple explanation (in the case you were interested to know ruling nerd litigation): The card has two requirements. 1. Ungloliant's Spawn existed since the card has been in effect. 2. Ungoliant's Spawn has been defeated since the card has been in effect.
You win the game if both are satisfied.
Card's not in play don't usually keep memory of games stats until it has been in effect. (unless otherwise stated)