Can you post a photo?

That looks like a decent size for eating and hanging out. Depends on what else you want to do out there. You could add on to it with a second level of decking. Or put in a patio at ground level, but the big difference in height would make it look a bit awkward if your idea is to have a larger area for entertaining.

Eating and grilling....what else is there?

All I want is to be able to have some outdoor armchairs and a fire pit, but it seems like it will be too awkward putting it in now smack in the middle of the backyard. Any suggestions on design of the patio below?

You can easily put in a fire pit and more seating! We have a covered deck on one side of our house too and it will lead down to a 20x20 patio of paving stones. We will have a conversation set there for seating but you could easily put a fire pit and chairs.

Enclose the under deck area with lattice.

Given the height above grade of the existing deck, 10 steps down seems like a long way to go if there is to be "flow" and connectivity. Half that distance seems workable for a space that would add comfortable furniture. To incorporate a fire pit area, you might need to consider gas in order to have a tame fire. Be sure to check what code allows. If you want a wood burning fire pit, it would probably be best to have a third level at grade. One would need a map and views of your overall back yard in order to see where it might fit in.