Cally's ability is quite explicit, key words being
"once skill cards played into a check have been revealed", ie when they are all sitting there face up including destiny. "you may choose one card to discard" ie she looks at all the face up cards sat in front of her and chooses the one she would like to discard (if any).
if you pass the search for home you *immediately* gain two distance completely out of band and unrelated to the jumping cycle. The distance gain is immediately effective and the fleet does not resolve a jump (ie nothing happens regarding ships or destination cards or the jump prep track). if the fleet was at distance two or three already this will immediately trigger the sleeper phase when resolving the pass effect (the same way you would if you passed legendary discovery crisis at distance 3).
The only reason it is not moved immediately to the stack of revealed destinations is it needs to keep blocking the mission card slot to prevent further missions till the next normal jump -however it *is* immediately counted.