I noticed when sleeping on a normal pillow, i did feel it pushing against the back of the unit. But I have memory foam pillow i hadn't used in ages (it's one of those ergonomic pillows shaped to support your neck). Anyway, I have a silk (i say silk it's some man made satin or something but feels like silk) pillowcase that i put on the support pillow and i no longer feel the unit being moved when i lie down.
Perhaps you are blaming the wrong thing, it may not be the front hairline you are having problems with at all but the back, and it is pushing the unit forward all night and causing the front hair line to lift up.
Check amazon too for nightcaps, there are loads of silk caps on there and cheap. I did toy with getting one but figured it may feel annoying which is why i just put a silk cover on my memory foam pillow and that seems to be enough.