I made a similar one in junior high shop. My teacher stressed thinking through the operations before making chips. I started with the butt end of the handle sticking out of the lathe, faced, drilled/tapped a 3/8"-16 hole. Then made an arbor (probably a 3/4" round) that threaded into the butt end of the hammer handle. The arbor could be chucked up so the rest of the handle could be worked. Center drilled the hammer end and turned the rest between centers. Made a brass cap to thread into the butt end hole.
Bruce