Perks up initially? Doubly right, there’s sign of contentment (4)
I believe the answer is:
purr
'sign of contentment' is the definition.
(I've seen this in another clue)
'perks up initially? doubly right' is the wordplay.
'initially?' suggests taking the first letters (I've seen 'initially' mean this).
'doubly' says to double the letters.
'right' becomes 'r' (common abbreviation).
The first letters of 'perks up' is 'pu'.
'r' duplicated is 'rr'.
'pu'+'rr'='PURR'
'there's' acts as a link.
(Other definitions for purr that I've seen before include "content? I might" , "Show pleasure" , "Sound of contentment" , "Express contentment" , "Happy feline sound" .)