Once you get your gear dialed in and you are sure the Hammock Life works for you, you can invest in something like the TensaOutdoor Solo pole. It's a collapsible aluminum pole that's sturdy enough to be a support for one end of a hammock (and tarp). You'll be trading the need to find two good trees with the need for two good ground anchors to guy out the pole.
Though it is natural to think of a tarp as rain protection, for me, it is mostly a sun/moon shade. Ultraliters may seek the smallest possible tarp that will do the job, which usually means just covering the hammock. I can handle the weight of a few more ounces and prefer a larger tarp, something that will cover the hammock and give me a dry area outside the hammock.