Constant advantage getting close to celebrity (6)
I believe the answer is:
steady
'constant' is the definition.
(synonyms)
I don't understand how the rest of the clue works.
Can you help me to learn more?
(Other definitions for steady that I've seen before include "Not faltering" , "Firm - sweetheart" , "Regular, constant, unvarying" , "Uniform" , "Not subject to change or variation" .)