Take back note with greeting to engineer (6)

I believe the answer is:

rehire

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'take back' is the definition.
I don't know anything about this answer so I cannot judge whether this works.

'note with greeting to engineer' is the wordplay.
'note' becomes 're' ().
'with' says to put letters next to each other.
'greeting' becomes 'hi' (hi is a kind of greeting).
'to engineer' becomes 're' (Royal Engineer).
're'+'hi'+'re'='REHIRE'

Can you help me to learn more?

(Another definition for rehire that I've seen is " Make employee over".)

I've seen this clue in The Hindu.

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