Multiple things to address so this might be longer than I like.
(1) DO NOT set the top of the handle against something and pound on the bottom of the head!!!!!!! With either a hammer or a punch, you WILL deform the bottom of the eye badly, and will NEVER get a handle to fit properly until you completely reshape the inside!!!!!! If you have a specially-made brass bushing to closely fit the handle and bottom of the eye, and a specially-made tube to pound on it with, you could avoid this problem. But then, you might as well use the arbor press with an equivalent bushing to support the bottom of the head as I do on production hawks.
(2) The traditional method is to use inertia to seat the head by banging the top of the handle on a block. As Cliff pointed out, there are problems with that method. One he didn't point out is that if you miss you've just trashed a handle. That's why I use the press. In the field, there isn't really a good alternative, but on a stump you're less likely to break off a chunk than getting "off target" and hitting the edge of the anvil.
(3) Rounding the ends (as Brian mentioned) is in my opinion an absolute NECESSITY to prevent sharp edges which WILL cause chunks of the handle to separate when they hit something (as in a poor throw). I have been thoroughly rounding handle "corners" for over 10 years.
(4) Brian's idea with the pickup bumper is not as far-fetched as it might sound, IF the hitch ball hole is big enough and IF you have a brass bushing to support the bottom of the eye ... it would be a lot like an upside-down handle press.
(5) Ryan's hot-formed punched eyes were a technique used by the best toolmakers 150 years ago. If carefully done, with very smooth forming mandrels, you'll get an eye about as clean as an investment casting which requires only a bit of filing and some abrasives to finish up very nicely. Unfortunately, most forged-head hawks use a wrapped eye and the "Rendezvous hawk" makers often don't put much care in trying to get the interior of the eye properly aligned, properly shaped, and relatively smooth. They are also frequently poorly hammer-welded where the blade reaches the eye, and they just run a TIG-weld bead up the point to try to keep it from splitting. RYAN DOESN'T DO THAT KIND OF SH*T, BY THE WAY! He is a maker of high quality hawks who takes care and pride in his work.
For good hammer welded "Rendezvous" hawks, look to H&B Forge. Their eyes are irregular enough that you'll have to spend quite a bit of time with the files to get a perfect teardrop eye, but their welds are rock-solid and they guarantee their heads. Their prices are not much higher than some pretty poor stuff you can find in the boxes or bins by the trade tables.
TWO HAWKS
http://www.2hawks.net