Yes the game does have a separate chapter that covers post-game content. And while some, if not most of these sidequests can be started during the main story, I just decided to wait and put them all here to keep myself from getting side-tracked too much.
Hammerhead Outpost
Let Sleeping Mountains Lie
This quest is triggered as soon as you return to past Lucis using Umbra's powers for the first time. Go speak to Cindy in Hammerhead and she'll tell you about the earthquake problem over at the Longwythe Area. Go find Dave next and he'll tell you to have a word with Dino. At this point, most of the NPCs (well those who have given you missions at least) you've met throughout the game will gather in Taka's diner. Talk to each one of them to get items. Speak with Dino and he'll send you to Monica. Go find Monica to unlock the Adamantoise hunt in Taka's diner. From here you can take on the Adamantoise at any time by accepting the hunt at the Hammerhead's diner. You don't have to kill the creature to complete this quest though. You just need to see the Adamantoise then flee from the scene.
Into Unknown Frontiers - Regalia Type F
For this mission, you need to first get you hands on the Strange Engine, an item you get as a reward for completing Formouth Garrison. Give Cindy the item once you have it so she can install it into your car. This will add flight capabilities to the Regalia which is great and all, but the thing is its landing parameters are not very precise, leading to crashes (and game overs) more often than not. Stick to the ground and drive instead. You have infinite fuel anyway.
Secullam Pass
The Jolly Joyrider - 350 EXP
Here we have another guy with a broken car that requires a Repair Kit yo fix. You'll find him at the parking spot in Secullam Pass by the way. Buy a repair kit from any item shop and hand it to the man to get this over with.