Climbing up on the structural ridgeline soapbox.
The Hennessy is designed to have a set amount of sag, determined by the structural ridgeline. Pulling the suspension too tight simply over stresses the ridgeline but doesn't change the lay of the hammock. Variations in the lay only occur once the ridgeline is less than taut.
Stepping off soap box.
As far as the wall of fabric on one side of your head I think that's fairly normal in most gathered end hammocks. When laying on your back with the standard head towards the right, feet left, the material to the right side of your face will rise up higher than the left edge.
In the pic below the person is laying on the opposite angle, but you can still see how the material is high on one side.
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