FBI agent, alternatively a gumshoe's hat? (6)
I believe the answer is:
fedora
'gumshoe's hat?' is the definition.
(hat a detective or 'gumshoe' might wear)
'fbi agent alternatively a' is the wordplay.
'fbi agent' becomes 'fed' (short for federal agent).
'alternatively' becomes 'or'.
'fed'+'or'+'a'='FEDORA'
(Other definitions for fedora that I've seen before include "Felt hat dented lengthways" , "Or fade away with man's soft felt hat" , "Felt hat, dented lengthwise" , "top fashion" , "Headgear" .)