There are several issues in play here that cannot be answered without a site-visit to examine the existing structure. I'd recommend getting at least three proposals and bids from local remodlers.

srecrcr - That's the current plan - a single level deck - but b/c of the slope of the land - it'll ending being higher than our fence line. We will be adding "screening" foliage but we were afraid that it woukd look ackward. DH thought the same thing about the grill, but it really wouldn't be that far. Here's what the rear elevation will look like (less the hardscape - patio etc.) There will be another set of french doors to the right of the deck (leading into the new morning room - where our kit. table & windows seats will be)

FWIW it is worth to those looking at the options like Localeater. Did some research today. A 10x14 pergola starts at about 4k (pine) and goes up based on material and features. We like Walpole (have bought from them before) and they make a nice one in Azek (looks like wood but wears better ... we used it for new steps at our beachhouse) for 10k. The retractable sun shades for the top are another 2k. 8 weeks delivery, alas. Then of course we need the mosquito curtains, another 1k or so. We would also need to dig up our stone patio to put in a foundation, then the put the patio back and drill through it. Meh. An awning with Sunbrella is about $3200 for a 9x16 (that size would nestle in nicely near our koi pond area). 6 weeks plus. More expensive if it is retractable, but that kind cannot accommodate mosquito nets. The awning vendor had a product that was like a screen that came down at a push of a button, hidden up under the awning. That was another $10k though. Alternatively, you can use mosquito curtains. Not crazy about any of these options so far ... we might start simple with mosquito curtains for our french doors off the MBR in Maine and see how the y work. I like that we can get them in white...

OP, you're showing us a perspective view. There's a limit to what we can understand with having only that view to look at. How wide, long and what shape/angles is it really? If we were a bird flying overhead, wouldn't the patio and house configuration be more obvious to us as to what we're looking at? Of course, it would! The PLAN view (which is like bird's-eye view) is the easiest way to understand landscape space -- its shape, size, orientation and how one part relates to the next. To START the design process, you don't need a FINISHED PLAN (though that would be the ultimate goal.) You only need a BASE PLAN, as a way to communicate to us what you have right now, before any work is done. The base plan would show the house footprint, the patio, the pergola posts and the nearby tree trunks. That's all. It is on a copy of this base plan that one would explore possibilities and be able to communicate them back to you. There's no good verbal way to do it. And because in the perspective view photo has some odd angles, it's a difficult way to do it, too. If I tried to draw the base plan for you, it's hard for me. I'd be wildly guessing about sizes, proportions and angles, so couldn't help but have a result that is inaccurate. But you're there so it's easy for you. Take a tape measure and measure the wall and patio lengths. Plot them out to scale on a piece of paper. It is literally a 20-minute process. By the orange line you drew on the picture, are you indicating that you want to expand the patio along the back wall of the house and you wish that part to be the covered part? If it's something else, it's not clear.