I initially felt the same as you but it went away quite soon. Try sleeping without the tarp or with it pitched only on one side....being able to see more open air helps. I own a Claytor Jungle hammock with a side zip, which would allow you to leave the zipper partially open. Doing some "mental gymnastics" may be required. I've done a ton of sleeping in my hammock this past summer. I actually slept over twelve hours straight the other night while backpacking....a record sleep for me. On a side note....I can see where the Hennesy with a bottom entry and a sewn bug net might accentuate any claustrophobic tendencies you might have. I'd definitely try a side zip. Also with the Claytor the netting is well away from your face which helps.
Good luck and don't give up....it's more than worth the effort, Miguel