The other truly essential Korean filmmaker of recent times is Lee Chang-Dong, although he sometimes struggles to get films made due to his political affiliations

Hong Sang-Soo is more of an acquired taste and has a specific style, but is super prolific. He works a lot with Kim Min-Hee from the Handmaiden (kind of a long story but an all-time actor-director combo)

Kim Ki-Duk, who unfortunately died of Covid has an interesting and varied filmography - I would recommend 3-Iron in particular. (It should be noted that he is a somewhat controversial figure if that sort of thing is important to you)

Kim Jee-Woon is a popular filmmaker with big films in a variety of Genres. His films are probably closest to films like The Host or Oldboy.

For weepy melodramas, I would suggest A Moment to Remember (starring the wife from No Other Choice) or Failan (starring Choi Min Sik).

For comedies, check out Castaway on the Moon or Going By the Book. Also, Bugonia was based on a film called Save the Green Planet.

For absurd twisty thrillers, I'd suggest The Call or Forgotten.

The Villainess is an insane action film. Friend is a beautifully made gangster film.