Yeah, either on new day or after entering one of the main buildings. There already were events like this. If mission stuff could be tied to this it would be great, because mostly it is dead end, and even early in game whet it was somewhat relevant it never was really fun. But I would be hesitant to try to ask creators to devise new mechanics. Problem with the missions is that Game is not building any goals, that making missions could be used to achieve. After heroines are captured, they never break away, their day-versions or they allies might as well not exist.
I really, really like SU, it is one of my favourites here. But some systems really lack purpose and connection to game as a whole. Even the whole boat event could be more fun if it had some apparent reason, like "information gathering" or "generating publicity" and make them parties not only for baddies. And could be done without drawing single new asset.