There are some very comfortable headlamps out there now, but if you really don't want to use one; UltraSpire makes a belt based light, Streamlight and Sofirn both make lanterns. First-Light has a belt mount for their angle flashlights. Pelican, Fox-Fury, Streamlight and Milwaukee all make scene lights, although we are really getting into overkill here.
If I had to pick a non-headlamp to use while chopping wood, I think a lantern would be my go to option. A light mounted below shoulder height is great for running, with the off-axis light giving good depth, it's going to make some weird shadows with chopping, and an axe/maul isn't something you want to be using while slightly disoriented.