Bees swarm when hive congestion is perceived (cells are filled, queen has no place to lay eggs). Splitting can prohibit this occurrence but only so far. Once began (queen cells) its pretty hard to discourage them from doing what is natural, reproducing or creating a swarm (think how hard it would be to stop a sexual encounter in the middle of the act and you'll understand the compulsion to swarm). Only regular inspections and proper manipulations can slow or stop them, especially during a good flow in June.
Providing room for your queen to lay eggs by removing and/or spreading out brood frames is another way, besides splitting. But you must be on top of it.
Simply adding supers with empty frames on top, as those below fill up sometimes works, best for honey supers. But for expanding the broodnest (preventing swarms) you need to do some frame manipulations. Removing 2-3 brood frames, replacing with empties and then placing those removed ones above along with more empties. Placement should be alternating with filled and empty frames, keeping them centered.
You can place honey frames to the 'sides' of the broodnest as long as they are the same size frames (you indicated DEEPS), but most of the honey should go right above the cluster for winter consumption. Bees move 'up' to feed throughout the majority of winter, often leaving honey on the sides behind and if little or no honey is above them they can starve despite ample honey below. So it goes….