A drift is usually cast ductile iron, but you could make one from steel. If you make one, do not harden it. You don't want to be striking hardened steel, and the hardening would be ruined the first time it was used, anyway. If you are beating your drift up to a point where it is unusable, the hawk head is most likely not hot enough.Get the head up to near welding heat to drift the eye and it will go quite smoothly. The drift's end will mushroom somewhat during use, and any sharp lip that forms can be ground away as needed. When you get the drift, it will have a rough cast surface and the casting seam may be sticking out. A quick cleanup of the seams on the belt grinder won't hurt. The rough surface on the rest of it is OK as long as there aren't any big bumps sticking out. No need to make it smooth all over.
You should also get/make some smaller drifting tools to open up the slot. An old hot punch ( looks like a hammer with a spike) can be re-shaped to make a great starter drift. You place the punch in/on the slot and strike the other end ( looks like the hammer head) with a maul.
You can make your own hot punch from a large ball peen hammer head. Draw the ball out into a 4" long spike and shape it to a wedge shape. Re-handle it and you are ready to go.
Now, I will give you another great idea.
There are two cardinal things in drifting a head -
1) Drifting a hawk/axe is really a two man job.
2) Everyone ends up burning themselves on the drift.......it gets hot fast.
A handle will help a lot in controlling the drift and keeping your hand un-burned and un-smashed. Take a 20" piece of 1/2" round mild steel stock and flatten about 6" of one end. Fold 3" of that back around and weld it to itself for about 2", leaving an unwelded slot of about 1". Heat it to forging temp and drift the slot until it goes about 2" from the end of your drift. Take the other end and bring 6" of it around to make an oval loop grip. Now you have a 10" long fitted drift handle. Besides keeping you hand from being burned, a big plus for this tool is that it is less nerve wracking to have a striker hit the drift for you when your hand isn't directly under the spot he is hitting.....especially when the striker is inexperienced and swinging a 4# maul.
Cost - $1-2. Money saved on burn cream and trips to the ER for wrist X-rays - $5-500. Value of making your own tools and using them - Priceless.
Lots of folks sell drifts. Blacksmith depot sells a good one, as do many other suppliers. Unless you make a lot of hawks, one drift should last for life.