New act with hat and stick (6)
I believe the answer is:
attach
'stick' is the definition.
(I know that stick is a more specific form of the action fasten)
'new act with hat' is the wordplay.
'new' is an anagram indicator.
'with' is a charade indicator (letters next to each other).
'act' put after 'hat' is 'hatact'.
'hatact' with letters rearranged gives 'ATTACH'.
'and' is the link.
(Other definitions for attach that I've seen before include "Associate" , "Join, connect" , "Tie, bind to" , "Stick" , "Bind, fasten" .)