Volunteers to rid the chests of their cold, to start with (6)
I believe the answer is:
offers
'volunteers' is the definition.
('offer' can be a synonym of 'volunteer')
I cannot understand how the remainder of the clue works.
Can you help me to learn more?
(Other definitions for offers that I've seen before include "Makes bid or tenders" , "Presents for acceptance or rejection" , "Tenders, presents" , "volunteers" , "Bids, tenders" .)