The same method works for making a handle for an adze, pickaxe, pulaski, or any tool where the head slides up the new handle and jams up on the large end of the handle.
Step 1: Get Wood
Don't forget firewood! Check out your woodpile, I bet there's some good hardwood out there that's well seasoned already. Green wood is okay for a pick handle also. Green wood is actually a lot faster to work with hand tools. It will shrink a bit, but that's not a problem, the big blob on the end will keep the head from flying off. Try your park department or trees blown down in a storm.
Orchards prune a tremendous amount of wood from their trees. They all have to pay people to get rid of their wood.
Hickory and its relatives are the strongest woods that grow in the U.S., which is why they're usually used for tool handles. Ash is almost as strong and I happen to have some. Don't use softwood, it just won't last, and you'll get to make another handle. On second thought, go ahead!
If you're in the south or in Mexico, hard yellow pine is plenty strong although it's officially a softwood. That's why they use it to make ladders and floorboards.
In warm areas like that invasive "ironwood" a.k.a. Casuarina, called "pine" in the Marshall Islands is great for tool handles. Shape it green and wet, it's very easy to work. Then throw it in salt water until it sinks. It'll turn very hard and a pretty red-orange color. The New Zealand Maori call it "Toa" or "warrior" because they make their fighting staves from it.
Here are some tool handles I made on Majuro Atoll. The axe handle and maul handles are ironwood.
I didn't make the small adze handle, but both adzes are cutoff pick axes. The rightmost maul head is breadfruit wood, lightweight so the neighborhood kids could help me split logs. The last photo is my pile of ironwood splitting wedges. Some of the wedges fell in the lagoon and when I found them again they'd turned into super-tough orange miracle material.