>>493777I'd be wary about a 4x4, but in a diamond configuration you could make it work under a hammock. It's going to weigh a bit more than other tarps obviously, but it will also won't compress nearly as much. I make a "bedroll" with my 6x8, rolling a wool blanket or a sleeping bag inside, and strap it to the bottom of my pack. The only issue I have with mine, no tie outs, it has grommets.
The key is to make sure the treatment on tarp works. Lightly pour some water on it and see if it pools. The $40 one I bought from Chicago Canvas had a decent treatment, but it began to leak after a little bit. By mixing some wax, linseed oil, and turpentine, you can make a great cheap treatment for canvas and cotton in general. I used the extra to treat my knives as well. Obviously keep it away from firearms.
Now, the main reason I went to canvas was because of the ground contact my tarp always had, causing fraying and tears. You may find that a synthetic tarp may be more practical, as when used with a hammock it doesn't touch the ground. Just my view on it, it may be heavier than it's worth for you.