Thanks for the welcome, everyone. Kinda weird, though--I've been reading this forum for almost a year and I feel like I know you all pretty well already!
Anyway, thanks, Javaman, for that description. Makes a lot of sense. I think, though, for your technique to work for me I'd need to move the rings onto the hammock suspension rope, instead of the treehuggers they are currently mounted on, but maybe I wouldn't need to do that--I have a pretty long reach. I went that way mainly to have as much adjustability as possible. I also tried to pull a bight through the rings, but the line twisted as I pulled and the kinks that formed would not go through the rings no matter what I did.
I'll have to hang it again and try it out. Won't get to do that again until Saturday, though.