Edward, here's what I was going to post yesterday. We are heading into the fall season, maybe the tail end of the "three seasons" you mentioned. Maybe the gear you have will work a little bit longer as you get a better idea of the options. Remember, if you are going to use that ground pad inside the hammock, don't inflate it as stiff as you would using it on the ground.
Here's my problem, when I bought my gear is was years ago. An underquilt protector cost $30. Now I've seen them for $100! So it's difficult for me to advise because a lot depends on your financial situation.
For me, I knew there were no "classrooms" for what I wanted to learn. So I'd buy gear, try it out, learn from it. Then, if I was ready to move on from it, I'd sell it at about 30% off the price I paid and chalk the dollar difference up to the "cost of education".
Anyway, "Here's the rest of the story ... "
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Edward, do understand that there is no "absolutely correct answer" and there are a lot of opinions. There are also marketing trends. For example, in another field, some kayaks from the UK started appearing with what was called a "Day Hatch" located behind and to the side of the cockpit where you sit. The problem was, you had to be a circus contortionist to twist and reach back to open it. But it was "new". Soon, many new sea kayaks from other vendors appeared with a day hatch. It was "the next new thing."
Along the same line, Hennessy had their classic, "no zippers" opening built into the hammock body. But everyone else's hammocks had a side zipper opening. So Hennessy came out with the zipper opening option. Doesn't make it better (or worse). But it's important to see what's going on.
Superior Gear came out with a hammock with an underquilt sewn to the body of the hammock. Now more and more vendors are offering the same option. Is it better? Maybe for some, maybe not for others. Think about it ... You could have one hammock and have UQ's for winter, spring/fall, and summer. Or you would need three Superior Gear hammocks, each with a different temperature range. If I want to wash my hammock for any reason, I put it in the washer (side loading, without a center agitator). With an attached UQ hammock, I have to consider the details of washing the down fill.
But, if I knew I was only going to use it for most of the summer and some shoulder season, and I normally keep my gear clean (I sleep in light jammies and socks - easier to wash clothes than to wash gear), then maybe a hammock with an attached UQ rated around 30° would be the cat's meow. Less pieces of gear. Always the correct fit.
You are absolutely correct in that bottom insulation is critical. There are a lot of things to consider, and it takes some experience. I'd rather have you wait a bit, then get a UQ and exclaim, "That darned cougarmeat suggested I wait. What took me so long! This UQ is great." then to spend money on something later feel like, "I could have had a V8." (old commercial references).
Here's an idea. Just an idea, not "the best" idea. You already have a sleeping bag and pad. Assuming it's rated for your adventure temperature range, you can put the pad in the hammock and maybe unzip the sleeping bag down to about 2 ft from the bottom (creating a "foot box") and use it as a top quilt. Note that, because it will be open at the sides, it won't be as warm as when it is fully zipped. But it might be warm enough.
This is the least expensive because you are using gear you already own. It will also be the least comfortable, but hopefully still more comfortable than ground sleeping.
Some people then move to a double-layer (DL) hammock. That hammock has an extra sleeve at the bottom that you slide your pad into. That way, it doesn't move around and is usually more comfortable than having the pad in the hammock itself. Also note, when you are using your pad like this, you are using it as an air block. So you don't need to pump it up so it is rigid; you want some flexibility to bend with your shape.. But your Hennessy is not DL. So ....
A next step is to get a UQP (underquilt protector). It's just a sheet of breathable nylon that uses a shock-cord to hang under your hammock body. Some people don't like them and think they are unnecessary. I have them for all my hammocks. You can put your pad in the UQP, which hangs under the hammock body a short distance, so you don't feel the pad at all. The sides of the UQP come up to the sides of the hammock, so you have a little chamber of non-moving air insulation. Plus, the UQP blocks air currents under the hammock.
Hennessy has their own "Super Shelter" - something like that - which is just a UQP with some additional foam you put in it. But you already have a pad so look for just a UQP. There was a company called 2QZQ that sold them for about $30. 2QZQ was acquired by DutchWare. I don't know what UQP's are going for these days.
Then you have the Gold Standard - an UnderQuilt. Though I have UQ's for 0°, 20°, and 40°, after a few years, it looks like - around where I adventure - I'd only need two, a 10° and a 30° to pretty much cover the same months. If I lived in a warmer place and skipped winter, a 40° would most likely do the job just fine. For me, that would be good for 50° nights and above. I know that because I've slept out at home and used a min/max ($10) thermometer to show me how cold it got at night.
Oh - there reason I use UQP's is I have a bunch of hammocks and UQ's that I swap around. If I had just one hammock and one UQ, I could set it up just right and not give it a second thought. But it's fun to play with different toys. Problem is, quantum physics aside, I haven't figured out how to be IN the hammock and OUTSIDE the hammock at the same time to adjust the UQ. The UQP allows me to be a little sloppier with my setup. It also keeps the UQ a little cleaner when the hammock slips out of your hand as you are putting it up and taking it down because the UQP is a barrier between the ground and the UQ. If it's raining, and you have a minimal tarp, you can get some "bounce" under that tarp, and the UQP again protects the UQ. Also, because of the extra wind blockage, it can extend the temperature rating of the UQ.
However ... Another person can say, "cougarmeat, you are all wet. I just rig my tarp (with doors) up close to the hammock. It keeps the hammock dry. It keeps an envelope of warmth. And I've learned not to drop my hammock during setup/takedown." And they'd be right too.