Explosive, overwhelming greeting offends volunteers right now (4,7)
I believe the answer is:
this instant
'right now' is the definition.
(this instant means at this very moment)
'explosive overwhelming greeting offends volunteers' is the wordplay.
'explosive' becomes 'tnt' (TNT is an example).
'overwhelming' indicates putting letters inside.
'greeting' becomes 'hi' (hi is a kind of greeting).
'offends' becomes 'sins' (offend can mean to commit a sin).
'volunteers' becomes 'TA' (abbreviation for Territorial Army).
'hi'+'sins'+'ta'='hisinsta'
'tnt' enclosing 'hisinsta' is 'THIS INSTANT'.
(Another definition for this instant that I've seen is " Right now".)