You may never explore on another player’s turn.
I think the question is in the language on the cards detect evil/magic. They don't say explore. They say encounter.
"If it is a *card type*....you may encounter...."
Is an encounter treated the same as exploring?
I don't believe so otherwise how do I "encounter" an ancient skeleton henchman that was summoned from someone else finding one on their turn. That card specifies to have an encounter during someone else's turn.
From that logic you should be able to cast detect magic on another players turn and encounter that card if it has the magic trait correct?