hudson_b, how long will you be out? Will you always be in "open space"? The major consensus is that an underquilt is the most comfortable option, but many just use their ground sleeping pad in the hammock (or in the hammock sleeve if it's a double-layer hammock). If that pad is inflatable, then you need not blow it up as firm as you would when using it on the ground; you want it more flexible so it shapes to the hammock curves.
There are a few considerations when choosing a UQ. What fill do you want? I started out with synthetic because of a "mist" experience on Vargas Island. But later moved to down because, for me, that "morning mist, everything was saturated wet" was a one-off, and with an adequate tarp and underquilt protector, I was sure I could keep my down dry. What temperature range do you want. There are some postings that recommend 40° rating for three seasons. That might work for three seasons in California, but maybe not in the Pacific Northwest - unless your shoulder seasons are short. Better to mention the temperature range you expect to be in. Bend is different than San Diego.
But if it is truly open space, a hammock isn't applicable everywhere. There are variations of portable stands, but they move you out of the "budget" category. You can set your hammock up as a bivi bag - put down a ground sheet, then pad, then hammock. Support your tarp with hiking poles (guyed out) and tie the hammock bugnet on those same poles, lifting it off your face.
But if that were the situation for a few days, I'd probably bring a tent instead. Sand is the worst environment for me. It gets in everything. And anchoring/guying tarps and poles is a challenge.
My suggestion of using a pad you might already have is so you know you don't have to be in a hurry to buy new gear. I would always recommend a UQ, but I would not advise some going out a dropping a couple of hundred dollars just to cover a 3-day camping trip. And there's a hesitancy on "budget" because instead of budget/new, you could get better quality/used. But finding deals and understanding the options (example: footbox - cinched, zipped, or sewn) is not something to do in a hurry.
