Any game is fine, even if the episodes are minimal, for example Dead of Winter Long Night (I believe) has 3 or so episodes. I have the Arkham Horror LCG and we might dive into that next, any other good episodic or story based games? (note, other than Gloomhaven as that's too much money right now)
Space Cadets: Away Missions are somewhat linked scenarios, but no character progression between them. So story, but not mechanisms.
Too Many Bones is sort of the opposite where the story is loosely linked with random encounter cards, but there is character progression from play to play
I think Arkham LCG is a good choice, but do want to mention that, being an LCG, you will very quickly be spending more total money on it than Gloomhaven and Gloomgaven is a one-time-cost with over 100 scenarios.