The lantern was always with him, from the time ate to when he went to bed. Though his hands weren't always on it, his eyes held it tightly, always making sure that it never vanished (as things in The Unknown often do). If it were gone, then the Woodsman would truly have nothing.
On some days, when he hauled home a new tree, the lantern burned bright, and from the flames he felt the warm, familiar presence of his daughter. When he had mere twigs, his daughter's flame was a weak flicker.
Still, either was better than the flame extinguishing.