Season in South Australia to worry a pet (7,3)

I believe the answer is:

sausage dog

I'm a little stuck... Click here to teach me more about this clue!

'a pet' is the definition.
The answer and definition can be both animals as well as being singular nouns.
Maybe you can see an association between them that I don't see?

'season in south australia to worry' is the wordplay.
'season' becomes 'age' (season can mean a period or age).
'in' indicates putting letters inside.
'south' becomes 's' (abbreviation).
'australia' becomes 'Aus' (abbreviation).
'to worry' becomes 'dog' (I've seen this before).
's'+'aus'+'dog'='sausdog'
'age' placed within 'sausdog' is 'SAUSAGE-DOG'.

Can you help me to learn more?

(Other definitions for sausage dog that I've seen before include "Dachshund (colloq.)" , "At a stretch, a pet" .)

I've seen this clue in The Everyman puzzle in The Observer.

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