Hat maker's factory in outskirts of Exeter (8)
I believe the answer is:
milliner
'hat maker's' is the definition.
(I've seen this before)
'factory in outskirts of exeter' is the wordplay.
'factory' becomes 'mill' ('mill' can be a synonym of 'factory').
'outskirts of' says to hollow out the word (remove centre letters).
'exeter' with its middle taken out is 'er'.
'mill'+'in'+'er'='MILLINER'
(Other definitions for milliner that I've seen before include "Garment maker" , "One making pork-pies" , "Ladies' hatter" , "Philip Treacy, say" , "Women's hat maker" .)