More stuffy person finishing (6)
I believe the answer is:
closer
This clue is a double definition.
'more stuffy' is the first definition.
('close' can be a synonym of 'stuffy')
'person finishing' is the second definition.
(someone closing something up)
(Other definitions for closer that I've seen before include "Nearer or more penny-pinching" , "more reticent" , "more precise" , "not so remote" , "Stuffier" .)