Hamlet's last run in southern resort progressed well (6)
I believe the answer is:
throve
'progressed well' is the definition.
(past tense of 'thrive')
'hamlet's last run in southern resort' is the wordplay.
'last' suggests the final letters.
'run' becomes 'r' (cricket abbreviation).
'in' indicates putting letters inside.
'southern resort' becomes 'Hove' (English seaside resort).
The final letter of 'hamlet' is 't'.
'r' going within 'hove' is 'hrove'.
't'+'hrove'='THROVE'
(Other definitions for throve that I've seen before include "Got on" , "prospered" .)