I assume your hammock is not that long and if not, like less than 11' I would just buy a
Wilderness Logics Tadpole. Good coverage and weighs 11.5 oz for $85.
I dont think you will beat that weight until you get into Cuben Fiber tarps.
I needed a longer tarp, so am building almost a duplicate only 12.5' long. Material cost alone is $55 from quest plus shipping plus a lot of work.
If you need a winter tarp, tackle that later.
That said Loner2012AT (dont know his name here) just did an AT thru hike with a tadpole tarp.