Dog’s bed originally inapt outside New Jersey (7)
I believe the answer is:
basenji
'dog's' is the definition.
(basenji is a kind of dog)
'bed originally inapt outside new jersey' is the wordplay.
'bed' becomes 'base' (I've seen this in other clues).
'originally' suggests taking the first letters.
'outside' is an insertion indicator (some letters go outside others).
'new jersey' becomes 'nj' (abbreviation).
The first letter of 'inapt' is 'i'.
'base'+'i'='basei'
'basei' enclosing 'nj' is 'BASENJI'.
(Other definitions for basenji that I've seen before include "Smallish hunting dog" , "Small hunting dog" , "Breed of African hunting dog that rarely barks" , "Dog unable to bark" , "Barkless dog" .)