Not talking about the Elder Rune Hatchet. But at some point, there is a chance for woodcutting ot get newer hatchets. Including the woodcutting equivalent to the Pickaxe of Earth and Song and Mattock of Space and Time.
I think it'll depend on whether the Inferno Adze ends up being used to make it. If it is, it'll probably be something like "Hatchet of Flame and Tinder". If not, it might be something like "Hatchet of Nature and Harmony".