The most common and most likely answer to this crossword clue is the 4 plus 3 plus 6 letter word LONG HOT SUMMER
L
O
N
G
H
O
T
S
U
M
M
E
R
Buddy explains!
The straight part
We know that "a period of unrest in the Sixties" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a period of unrest in the sixties = long hot summer.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.