Part of garment Oscar fed to pet dog (8)

I believe the answer is:

coattail

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'part of garment' is the definition.
'coattail' can be an answer for 'garment' (thesaurus). I'm not sure about the 'part of' bit.

'oscar fed to pet dog' is the wordplay.
'oscar' becomes 'O' (phonetic alphabet: alpha, bravo, charlie etc.).
'fed to' means one lot of letters goes inside another.
'pet' becomes 'cat' (cats are often kept as pets).
'dog' becomes 'tail' (synonyms).
'o' inserted into 'cat' is 'coat'.
'coat'+'tail'='COATTAIL'

Can you help me to learn more?

I've seen this clue in The Independent.

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