I was trying to find the same thing and haibane renmei is the perfect thing to watch if you like Lain, it is AMAZING you will love it if you love lain
i heard NieA Under 7 has the same vibe but not psychological i think, its more down to earth.
Inaka Isha is weird and psychological but its a movie with bizarre animation.
and of course evangelion, the movie, and the remakes, its not exactly what you want but its an AMAZING dive into grey philosophical psychological themes
some manga you might like is
"The climber" its psychological but not horror-ish its more about finding yourself, with GOD tier art
and story, the character development is insane.
"Goodnight punpun" its a psychological coming of age story, its amazing but not the vibe you are looking for.
"Biblomania" is a short read but an amazing horror, dark fantasy psychological story, it gets you hooked for its 12 chapters.