Chief constable's last state (5)
I believe the answer is:
maine
'state' is the definition.
(US state)
'chief constable's last' is the wordplay.
'chief' becomes 'main' (synonyms).
'last' indicates one should take the final letters.
The final letter of 'constable' is 'e'.
'main'+'e'='MAINE'
(Other definitions for maine that I've seen before include "North-east American state" , "Part of America" , "N. US Atlantic state" , "somewhere in America" , "NE US state" .)