Carefree Oscar called out "goodbye" (7)
I believe the answer is:
cheerio
'goodbye' is the definition.
(synonyms)
I can't explain the rest of the clue.
Can you help me to learn more?
(Other definitions for cheerio that I've seen before include "Farewell - drinking toast" , "Expression used when leaving" , "Say this before drinking" , "Ta-ta" , "A final word" .)