My other problem is that in summer, I don't need insulation under my hammock, and often I'll just use a silk sleeping bag liner (it's often 70-80 degrees at night in Florida in the summer). I'm protected from mosquitoes above by the netting on my hammock, but they bite right through the bottom.
I think a lot of folks have gone to double-layered hammocks for this. Personally, for weight savings and maximum breathability, I prefer a single layer hammock. I'm about to start a DIY bug net project that will basically hang from my structural ridgeline, droop around my hammock, and have a closure far enough below the bottom of my hammock that the bugs can't get to me. Like a big 'envelope' of netting. Similar to the ENO "Guardian", but a home-made version that's tailored a little better to my particular dimensions.

I'll probably just make it like a long tube (tapered at each end), with drawstrings at each end for closure. It's just a matter of installing it at one end before you get in for the night. After you get in, simply slide it up around you to the other end of the hammock, and pull the drawstring closed. I may think about doing a long zipper closure on the side for convenience too, but I've heard that takes some skill when sewing on mesh.

Alternatively, if you want to keep the setup you have, just spray some permethrin on the bottom of your hammock. The wee beasties will leave you alone.