Very interesting one because a lot of people do not want to see them broken up. I'll play along, for fun.
The breakup: I'm ultimately thinking would be "jealousy" but I want it implemented by another tag team. That tag team could have been AJ/Jericho because Jericho is awesome in that role, but I want it to be Enzo/Cass. Let Enzo/Cass stoop to the level of trash talk that eventually plants the seed for jealousy.
Maybe talk up that Big E gets the pin and that the other two are just riding his coat-tails all the way to glory. That sets Big E up to look like a monster and the other two can get jealous between one another. Then you can have Woods/Kofi fight for the #2 spot instead of someone on the outside.
That leads to the inevitable promo where they start bickering and ultimately to a ladder match against Enzo/Cass at a PPV. Enzo/Cass/Carmella can all be down and the 3 get in the ring to hold the ladder but they bicker over which ones grabs the titles. They start fighting and Enzo/Cass come through and get their big win. Enzo/Cass take over the belts and we get the split of the New Day.
The night after the PPV, have Big E walk out on Woods/Kofi to start his new main event run and have Woods/Kofi start pushing one another.
Post New Day:
Big E: I think it's time we see a main event run for Big E. Dude is crazy agile and big. I can't imagine he wouldn't be a face people would cheer for. He's got great charisma.
Kofi/Woods: Turn this into a feud.