Maybe went down the pub, then dropped round (7)

'maybe went down' is the definition.
(shin can mean to climb up or down a tree)

'pub then dropped round' is the wordplay.
'pub' becomes 'inn' (an inn is a type of pub).
'then dropped' becomes 'shed' (shedding is a kind of dropping. I am not sure about the 'then' bit.).
'round' indicates putting letters inside.
'inn' placed into 'shed' is 'SHINNED'.

'the' acts as a link.