I'll exclude ones previously mentioned
Made in the 60s
Branded to Kill
2001: A Space Odyssey
Night of the Living Dead
Yellow Submarine (The Beatles animated movie)
Barbarella
Batman (the Adam West one)
King Kong vs Godzilla
2001, and Barbarella aren't set in the 60s, but have so much of that time in them that I have to count them.
After the 60s:
Inherent Vice (Okay, Summer of 1970, but close enough, especially since it's set in LA)
One Flew Over the Cuckoo's nest (barely counts, based on a 1962 book, but made in the 70s)
Honestly can't think of many movies about or set in the 60s.
In the second category, American Graffiti, set in 1962 and supposedly inspired by writer-director George Lucas' own life in that period.
woodstock
yellow submarine
woodstock
yellow submarine
Breakfast at Tiffany's (set in 1960; released 1961)
Gidget movies (set and released in 1960's)
The Pink Panther (probably set in 1963; released 1963)
A Hard Day's Night (1964 for both)
* also other Beatles and Rolling Stones (Gimme Shelter) movies
Goldfinger (set in 1964; released 1964)
* and the other 60's Sean Connery/James Bond movies released during that time
Beach Blanket Bingo (probably set in mid-60's; released 1965)
* several other beach movies
The Graduate (set in the 1960's, probably 1966, since the Alfa Romeo he received as a graduation gift was 1966; released 1967)
American Graffiti (set in 1962; released 1973)
Apocalypse Now (set in late 60's; released 1979)
Platoon (set in 1967; released 1986)
JFK (set in 1963; released 1991)
The Doors (mostly set in 1960's/JM died 1971; released 1991)
Malcolm X (early 60's, assassinated 1965; released 1992)
Parts of Forrest Gump (covers entire decade; released 1994)
Once Upon A Time... In Hollywood (set in 1969; released 2019)
Can't believe we forgot the Bond movies. Dr No, Russia With Love, Thunderball, You Only Live Twice, Goldfinger and Majesty's Secret Service. Those are some quintessential 60's flicks.