American native given greeting outside work (4)
I believe the answer is:
hopi
'american native' is the definition.
'greeting outside work' is the wordplay.
'greeting' becomes 'hi' (hi is a kind of greeting).
'outside' means one lot of letters goes inside another.
'work' becomes 'op' (abbreviation for opus).
'hi' going around 'op' is 'HOPI'.
'given' acts as a link.
(Other definitions for hopi that I've seen before include "Pueblo tribe member" , "Arizona inhabitant" , "Member of a Pueblo Indian people of northeastern Arizona" , "Language" , "Arizonan Pueblo Indian" .)