Insurgent takes the French name again (7)

I believe the answer is:

relabel

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'insurgent takes the french name again' is the definition.
I can't judge whether this defines the answer.

'insurgent takes the french' is the wordplay.
'insurgent' becomes 'rebel' (synonyms).
'takes' means one lot of letters goes inside another.
'the french' becomes 'la' ('the' in French).
'rebel' going around 'la' is 'RELABEL'.

Can you help me to learn more?

(Another definition for relabel that I've seen is " Put a new tag".)

I've seen this clue in The Daily Express.

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