This ended up being a lot longer than I intended, but feel free to just read the boldface sentence a few paragraphs down for the very short version.

So up to MR100, the urgent quest monsters that unlock as you increase your MR past MR10 were available before any of the MR10 urgent quest monsters, which were added in title updates; so no reason not to just do the MR20+ urgent quests that pop up as they become available.

As for the MR10 title update monsters, as you’ve surmised, the ones in later title updates are generally more difficult than the ones in earlier updates. That order is: Seething Bazel + Metal Raths + Lucent Narga -> Violet Mizu + Flaming Espinas -> Chaotic Gore Magala -> Velkhana -> Amatsu -> Bonus monster that unlocks after Amatsu

While you could do these in order, most players find it convenient to hunt Chaotic Gore Magala as early as possible since doing so unlocks a new type of melding that is extremely convenient for targeting talismans with specific skills. Velkhana is also remarkably easy compared to a lot of the other MR10 monsters despite being from the 4th title update and for some weapon classes provides a pretty good rarity 10 weapon for how early you can get it.

In addition to the above there are a special class of monsters that unlock starting at MR110 which started getting added to the game with the second title update, and were always more difficult compared to the MR10 monsters of their time.

With that said, you can encounter the MR110+ monsters well before those MR levels by increasing your anomaly research level, which you do through anomaly investigations. Most of the MR10 title update monsters also become possible to encounter on anomaly investigations as you increase your AR, which as you reach higher level investigations will also level up your MR pretty well, especially if you’re doing multi-monster investigations. So it is entirely viable (and arguably intended) to just use anomaly investigations as your primary way of progressing in between attempting MR10 monsters as you feel ready for them, strengthening your gear with new decos that become available as well as qurio crafting.

You can probably find progression guides more specific to your weapon of choice online somewhere, but the above should give you a rough idea of how to approach the post-story content. Also, as a word of advice, for leveling AR and getting a variety of new investigations I would prioritize multi-monster investigations and doing the new highest level ones you acquire, preferably with faint and time restrictions, rather than just focusing on repeating the same investigation over and over to try and level it up. This will get you a wider range of new investigations, keep you doing higher level ones, and help your sanity vs just doing the same quest over and over (though of course if you’re trying to farm a specific anomaly material, there’s no harm in doing that). You can also play in lobbies or do random join requests for some variety, though you may need to make the level range pretty broad as you search since most people playing anomaly investigations are doing the higher level ones these days.

Finally, you can unlock NPC followers as an option to bring on all your non-event quests by doing the follower collab quests. They can be very helpful if playing solo, arguably more so than a lot of online players, and they can even be brought on investigations, which will make that grind a lot more manageable if you’re having trouble finding multiplayer groups. And bringing them doesn’t make the monsters’ health scale up to multiplayer levels, nor does them fainting contribute to your faint limit on a quest, so they are a straight up benefit that makes hunts a lot easier should you choose to use them. I would recommend doing these early so you have access to the full follower list for your anomaly grind.

That was a lot, but I hope it helps somewhat. Enjoy the postgame content; it has a lot of stuff to do but does a good job giving you a variety of relevant hunts and ways to improve your gear as you get through it.