I use a "sheath" that is on a thong that goes over the shoulder and the Hawk handle tucks in the sash with a small thong of leather holding the hawk to the sheath which is easily untied with one hand and then the hawk lifts out leaving the sheath to hang from the over the neck thong, I have never tried going thru brush and steep rough country with a hawk carried with an exposed blade and I don't think I care to try.I have a small bag axe that hangs from a protective sheath that is attatched to the strap of my shoulder bag, which is a canvas bag that looks verey much like a military bag called a haversack.
I guess the big question on all this is how sharp one keeps the edge of the Hawk. If it is not sharp and has a 60 degree bevel- sticking it in the belt/sash is probably okay but if it is so sharp you can shave with it, it seems having that exposed edge is inviting danger. It seems there is documentation to carry in either mode as there are sheaths of various sorts that were used.