My little daughter also buries her face either in the pillow or my chest and rubs vigorously even when asleep, to the point of waking herself up. Swaddling helps but restricting her also seems to wake her up as she tries to fight to free her hands.
Paediatrics is part of my course and I picked up that she had acid reflux when she was about 2 months old. Symptoms being: spitting up, pulling her face when burping, crying after a burp and not wanting to eat as
much.
After some research, I found a correlation between the silent reflux and face rubbing. I think any parent would pick up on face rubbing when tired and vigorous face rubbing. I tried all the methods of soothing her e.g. holding upright for 30mins, making sure she burps even if it takes an hour, feeding her in an upright position, giving the pacifier right after feeding and also I got these natural drops called Iberogast from my paediatrician. I give her these drops just before a feed.
All of this seemed to work. She's a smiling little 3 month old now.
She also has atopic eczema, so I make sure I put on her cream from her Paed once a day. I never changed detergent etc as this cream helped her.
Also don't forget about over stimulating baby. Reflux / fussy babies get over stimulated quickly. Monitor the times she's awake and look for signs of tiredness / over stimulation. (Random hiccups not related to feeding, fussing, yawning, not willing to interact or not interested in toys).
Soothing, in my opinion, doesn't last very long and they should eventually settle down without waking themselves up again to sooth. So just rule out all possible causes you suspect (in my case: reflux,thrush,eczema) and go with your instinct. Good luck to all!