I've just finished the game, and I have doubts about how the post game mechanics work, because this doesn't make sense.
After you finish the game and credits roll, the game proposes you to save and that save is named as "chapter 15…", however when I load that it is a previous save before even fighting Ifrit.
So.. how does that work? where is the game++?. New game is apparently a completely new game, I've tried that just in case.
Some people mention Umbra and going to the past or something, but at that point Umbra is not available, and my main quest is still about going to the Citadel and defeating Ardyn, which I already did.
Best Answer
You can call Umbra at any rest spot. There is a rest spot right before the last boss (I think to the left), and as such, you can use it to call Umbra and travel back to the open world mode.
There is no New Game+ (the story doesn't reset), you just keep doing Hunts and quests you didn't do in the open world.
Patch 1.03, released December 22nd, 2016 brings a New Game+ mode, where you can reset the story and keep your levels / items.