The best kind are styles that are designed to fit inside the sleeping bag such as the Kylmit Inertia X Frame. One of the biggest annoyances when trying to use a sleeping pad with a hammock is staying on top of the pad. It’s easy to shift your weight and move the sleeping pad from out underneath you. Some hammocks feature two layers to hold sleeping pads in place. Other hammockers like to stuff their pad inside their sleeping bag as long as it fits.

Alison hammock camping cold weather. And with a solid top cover hammock like this Dutchware Chameleon you can skip the weight and complexity of a tarp. I’ve comfortably slept down to around 10° F in a 3 lb (1.3 kg) hammock setup (hammock, top quilt, under quilt, tarp and suspension). That’s way lighter than most tent, sleeping bag, ground pad setups! [Note: a +20 under-quilt is not in the picture to better show the hammock body details.]

My other favorite is an old army surplus extreme cold weather sleeping bag. I was introduced to this bag by a friend. It weighs in around 10 lbs but that doesn't bother me. The bag is so effective, that I've found I don't need any other insulation below me to cut the wind or chill. I can just throw the darn thing inside my hammock, crawl in, and be really warm in a matter of seconds. Not sure what type of insulation is used inside the old surplus bag, but it is heavenly! It uses a zipper and snap system down the middle instead of down one side, like most bags. Makes crawling into the bag MUCH easier.Sleeping bags can also be used as top insulation to help cut the chill.

Leave the Asym tarp (napkin) at home. The majority of water is kept divorced from precious down by a trap with adequate coverage. Hex or winter tarps provide the most coverage and are easier to center over hammocks during setup. This margin in setup avoids multiple adjustments.  Also consider bringing a section of material to place under the tarp to keep gear clean and dry. One thru-hiker favorite budget option is a sheet of Tyvek.
Though down quilts are superior in many respects, there are also synthetic versions available. Like sleeping bags, these share the same advantages and disadvantages. Synthetic loft quilts are cheaper than their down counterparts. Unfortunately they are also heavier and do not compress as well. One advantage is that synthetic materials are resistant to the delofting effects of moisture. They keep their insulating properties even when wet. This is not true for down which ends up being completely useless if it absorbs too much water. A synthetic under quilt will cost more than a foam pad, but it can be cheap enough for budget campers. Keep in mind that quilts can only be used with a hammock. If for whatever reason, you end up sleeping on the ground, only a pad will provide your insulation!

Of course, always try to keep your hammock and your hammock straps as much balanced as possible. However, this doesn’t prevent you from hanging stuff on your straps instead of leaving it on the ground. It can be a camping light, your backpack, something you need to dry out. These parts of the gear can turn out to be very useful too, so don’t ignore them.
Starting my adventures in spring, I have a few months before actually having to invest in those quilts. I wanted to start with a second setup, for 3-season camping (>+10°C). Not exactly sure what to pack here yet. I was hoping a sleeping bag (or quilt) plus sleeping pad would be enough, but many of the not-exactly-cheap underquilts you tested were only just suited for exactly that kind of temperatures, so how could a sleeping pad suffice?
When you’ve got a hammock with you, you’ve also got a comfortable camp chair with you at all times, a place to sit and cook your food if necessary, hang out and tell jokes around the campfire, or just to relax and read a book while the day floats by. Not only that, it’s a hammock/chair you can take with you on your day hikes and set up next to that waterfall or after you’ve exhausted yourself on a climb.