Couldn’t you just mix the concrete right there on the remaining patio section, so you’d only have to shovel/move the mixed material over a few feet into the hole? Put up some sort of frame, on top of a piece of sheeting or just right on the old concrete, you could mix a lot all at once with just a hoe, then scrape or shovel in. Using slow cure concrete, you might be able to do all at once with two people, or maybe in sections just by yourself.