I would extend the entire width out

Florida or Hawaii or?

Extend a cement slab out the full width of the back patio as far as you need or the code allows. You can do a flooring of your choice. Think about a roof and screening to cover this extension and enclose the patio. Need that?

That is a magnificent privacy "hedge". :)

I'd do the full width as well. A slab would be better than pavers of course. If you can plan a redesign of all the rocky areas to get done at the same time, it'll probably save a little money as well as looking more cohesive, especially if you do anything like brickwork. Walks to the gates would be nice addition.

Extend the cement patio the full length. You may not need anything above it since you have a fairly deep recess in the ceiling alcove. Then use pavers or large pieces of cement if you need a walkway around the side.

Very private backyard enclave · More Info

@Christopher CNC It's Florida! Thank you for your help!