First thing after dinner, I've ended up saying goodbye (5)
I believe the answer is:
adieu
'goodbye' is the definition.
(synonyms)
I don't understand how the remainder of the clue works.
Can you help me to learn more?
(Other definitions for adieu that I've seen before include "Farewell (French)" , "Farewel" , "Vale" , "Bye" , "the last word in France" .)