Go after one inside the carriage (6)
I believe the answer is:
chaise
'the carriage' is the definition.
(chaise is a kind of carriage)
'go after one inside' is the wordplay.
'go after' becomes 'chase' (synonyms).
'one' becomes 'i' (Roman numeral).
'inside' indicates putting letters inside.
'chase' going around 'i' is 'CHAISE'.
(Other definitions for chaise that I've seen before include "... longue, chair for reclining" , "Light open carriage" .)