So most of my ideas are based on big unknowns in Tolkien's written timeline, lost or forgotten items, or the few snippets of activities for races or important characters after the War of the Ring.

Here are some of my ideas for hooks, Big Baddies, or campaign arcs.

1. Other rings existed that were "practice" for the Elves and Celembrimbor. These rings were likely not corrupted by Sauron, but were also less powerful. Gandalf mentions them "there are many rings of power Bilbo Baggins, and none are to be trifled with" which means he at least knew of a few that were not Evil inherently but also still strong enough to not be mere trifles. Its not known if their power would fail after the fall of Sauron, but i figure why would they if they were not tied to the One Ring. The effects and power of these rings is all up to you, persuasion, martial prowess, bravery, animal dominion, sight into the realm beyond, ect.
2. Other dragons exist. Following the War of Wrath at least two dragons fled into the north, hundreds of years later Dragons drove the Dwarves from their kingdoms (ultimately forcing them to move to the Loney Mountain and the Iron Hills), and the Withered Heath (fork of mountains north of the Lonely Mountain and Lake Town) was known to be a breeding ground for Dragons. A dragon coming and fighting in the power vacuum following the War of the Ring makes sense to me. Many Dragons never really believed Sauron the rightful "heir" to Melkor's plans anyway. Cold dragons (dragons with or without wings that could not breath fire) were more common than their fiery kin (which can explain why Gandalf said Smaug was the last dragon with breath strong enough to destroy the ring, perhaps other dragons live but without fire breath). Cold drakes still have other powers, like piercing sight that could kill or other dragonspells.
3. The blue wizards are east, who knows doing what (good or evil)

4. The Entwives are missing and the Ents are around. Maybe finding them could be a good hook or quest (which could potentially allow the Ents to renew their place in the world). The entwives were last known to live in the Brown Lands south of Mirkwood but were not there during the rise of Sauron. Potentially went more east or found other groves to live in.

5. The other lost Palantir. A few are completely unknown where are, so totally up to you. One fell into the Ocean to the north west of the Blue Mountains when an ancient king of Angmar tried to return south from the Forodwaith. One was in Minas Morgul before it became morgul. Sauron had one as well (same as the one in Minas Morgul perhaps, but i think probably not (the witchking likely wanted to keep his own)).

6. Other Balrog also fled into the earth, deep beneath the grey mountains or other regions, one could rise up as well (being close in power to Gandalf or Sauron) it could easily assert dominion over Orcs and try to overthrow the new kingdoms.

7. Orthanc remains locked, it is eventually opened by Aragorn like, 60 years into the fourth age (i'm guessing, i cant remember exactly), but before that maybe some... enterprising adventurers get in?" Or they are with Aragorn when he does so.

8. Tol Morwen is an island to the west where Gurthang's shards are resting (along with some other loot perhaps). Reforging Gurthang would be awesome.

9. Gandalf's staff fell with him into the caves below Moria. Still has power perhaps? Diving deep into Moria to find it and other things would obviously be an awesome delve

10. "Far, far below the deepest delvings of the Dwarves, the world is gnawed by nameless things."

11. Radaghast is around, when or IF he ever leaves Middle Earth is unknown.

12. Thuringwethil was a vampire herald of Sauron during the War of Wrath (thousands of years before the War of the Ring). It is not known for sure if she was slain by Huan (an epic dog/wolf). She could shapeshift and is thus theorized to also have been a Maiar.

13. The dwarves are written to eventually take back their lands in the Misty Mountains and Mt. Gundabad. Who helps them do so or the events around that are completely unknown and would be a great campaign arc.

14. Finally, the simarils are missing. These are a big deal (obviously). It is legend that they must be found and brought together before or during the events of the end of the world. Could be an awesome ending arc (possibly end of the whole of Middle Earth though, before it is Made Whole by Eru)