Chilly grandmother’s first to wear a hat (5)

'chilly' is the definition.
(I've seen this in another clue)

'grandmother's first to wear a hat' is the wordplay.
'grandmother's first' becomes 'g' (1st letter of 'grandmother').
'to wear' means one lot of letters goes inside another (to wear something is to be covered up by it).
'hat' becomes 'lid' (synonyms).
'a'+'lid'='alid'
'g' placed into 'alid' is 'ALGID'.