Newer CS hawks may have a set screw, but some of the older ones, like mine, i bought 10 or 15 years ago don't. But if you could drill the right sized hole completely through the head and handle(warning the metal might be alittle hard) you could put drive in a roll pin and grind the ends off flush withthe sides.
using a Wedge for an axe or hatchet would not help they are made for handles that come through the bottom of the head and then spread the handle and keep it from backing out. if you tryied this on a hawk handle it may just help to force the head down the handle.
If you remove the head from the handle and file the burr out from the inside edges of the head you will get more contact and a better fit, this really helps a lot.
when you put the head back on the handle use a deadblow hammer, rubber hammer, or block of wood er something like that to tap on the head end of the handle to force it on the handle for a tighter fit.
if you split the handle trying to use a wedge in it you might need a new handle.
hope these ideas help you out, Be Safe, :thumbup: Pat