What happens to overlay cards after failing dream?

Do overlay cards that were revealed and placed within a dream or slumber count as "gained" cards that are retained somehow and automatically re-placed at the start of the next attempt (when the underlying card is placed), or do they go back in the secret cards box and have to be found again on subsequent dream attempts?

(Not positive if overlay cards (not to be confused with quirk fate cards) exist for slumber too.... It's all becoming a blur) 😏

Thanks.