了 closes the act and lets it go (he put the hat on, and that's that). 著 keeps the result attached and present right now.
✗ 他在戴帽子 (for: the hat is already on) → ✓ 他戴著帽子
✗ 他戴帽子著 → ✓ 他戴著帽子
✗ 他戴了著帽子 → ✓ 他戴著帽子
English maps both the progressive (-ing) and the worn state onto one form, so learners reach for 在 when the result is already on the doer; 著 marks the worn state, 在 the act in motion.