At K’s Tavern and the Hunter’s Association in Shibuya, you can find a set of new quests to take on if you’re willing. A good few could have already been effectively done with proper exploration earlier.
"I Require Ample Weaponry:"
For this, gather the "Firearm Box" item. See Starting Investigation, the fifth paragraph under "Tokyo Overworld Map", for more details.
It’s basically to the north of the area. Note that it may be found after a randomized Domain. I initially didn’t get one before exploring Shibuya, and did after receiving this request and exploring randomly for the Tokyo Dome.
"Help Me Get Some Servers:"
For this, gather the Small Server item. Go to Hunt for the Shinjuku Underground and follow along the overworld walkthrough.
"Banquet of Flames:"
For this, head to the Front Entrance area where you first visited the main portion of Shibuya. In the southeastern corner alcove is a statue you can look at and call upon Cerberus. You’ll then end up fighting four Hell Hordes in consecutive order.
They are weak to Ice, repel Fire, and nullify Light/Dark. Stuff like Mabufula, Mabufu Stones, and Ice Breath are great here, as are Critical Wave and Megaton Press otherwise. They have about 1,000 HP.
After that first battle, it’ll go an area to the west, then deep into the Dogenzaka area. (You’ll go there later on.) Then head to the West Entrance area, two areas prior to Dogenzaka’s other half that you had to visit. In this final fight, he’ll have more like 2,200 or so HP.
Note that the end result can lower your alignment by one (first option) or raise it (second).
"Hunter Tournament Finals:"
The area needed is found to the north of the Tokyo Station near Kasumigaseki. Go north towards the river, as if you were going to use the guide. Go east around the river. Due north of where the ferryman/Hunter is is where you’ll find the battle arena.
Sadly, this mission delegates itself to a slaughter if you have moves like Critical Wave or Titanomachia party-wide, plus some elemental strikes.
Note that killing your foe will lower your alignment by one, and sparing them raises it by one.
"Training Battle 5:"
In this order, use Tetrakarn, Spirit Drain, Tetrakarn, Spirit Drain, Salvation, Spirit Drain, Tetrakarn, Spirit Drain, Salvation, Spirit Drain, and Tetrakarn.
"Training Battle 6:"
Have Flynn and the Gnome use Next, then swap Undine with Taotie, who also uses Next. Make the Salamander swap with Kanbari, and use Next until turn ends. On Turn 2, use Next until you get to Kanbari, who swaps with Undine. Use Next until the next turn.
On Turn 3, make Flynn summon in the Salamander and everyone else use Next. Finally, have Flynn summon the Kanbari, make Salamander use Next, and the Kanbari use Tetrakarn.