Need help improving a one month itinerary

We’re a couple, both 28, who enjoy both vibrant cities and beautiful nature. Here’s our current South Korea itinerary, and we’d love your feedback and suggestions for improvement: • Sept 18 – Land at ICN (Incheon) • Sept 18–23 – Seoul • Sept 23–27 – Rent a car and visit Chiaksan, Odaesan, and Seoraksan (likely one or two of them – any recommendations on which are best?) • Sept 27 – Return to Seoul and fly to Jeju • Sept 27–Oct 2 – Jeju Island • Oct 2 – Fly from Jeju to Busan or Pohang, then travel to Gyeongju • Oct 2–4 – Gyeongju • Oct 4 – Travel to Busan • Oct 4–9 – Busan (this will be during the Chuseok holidays – should we stay in a big city or escape somewhere else?) • Oct 9 – Travel to Jinju • Oct 9–11 – Jinju (Lantern Festival) • Oct 11 – Travel to Seoul (either slow direct train or via Busan with a high-speed train to Seoul) • Oct 11–15 – Seoul

Our return flight is on Oct 16, but we have one extra day to place somewhere in the itinerary – any ideas where it would fit best?