A love affair that's right out of the mail (5)
I believe the answer is:
amour
'a love affair' is the definition.
(I've seen this before)
'right out of the mail' is the wordplay.
'right' becomes 'r' (common abbreviation).
'out of' is a deletion indicator.
'the mail' becomes 'armour' (as in chainmail).
'armour' with 'r' taken out is 'AMOUR'.
'that's' is the link.
(Other definitions for amour that I've seen before include "A secret love affair in France" , "Love affair - loved one" , "Romantic affair" , "(Illicit) sexual relationship" , "Result of Cupid's darts" .)