This covering could make tool quiet at first (5)
I believe the answer is:
shawl
'this covering' is the definition.
(shawl is a kind of covering)
'tool quiet at first' is the wordplay.
'tool' becomes 'awl' (awl is a kind of tool).
'quiet' becomes 'sh' (both can be said to tell someone to be quiet).
'at first' says to put letters next to each other.
'awl' after 'sh' is 'SHAWL'.
'could make' is the link.
(Other definitions for shawl that I've seen before include "Cloth wrapped over one's shoulders" , "Piece of fabric wrapping the neck or a baby" , "Cover for head and shoulders" , "Afghan" , "Baby wrapper" .)