Can I use Close Encounter after my first turn, or only during my very first turn of the game?
You can use it on any turn, including your turn if you go first (when you normally would not be allowed to attack).
Source: Roaring Skies FAQ (2015-05-07), TPCi Rules Team (2015-05-07)

Relevant cards

Note: All applicable cards may not be listed below.
XY6_33