Wild pet housed in American region close to Mexico (4,5)

I believe the answer is:

deep south

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

'american region close to mexico' is the definition.
I can't tell whether this defines the answer.

'wild pet housed' is the wordplay.
'wild' indicates an anagram.
'pet'+'housed'='pethoused'
'pethoused' with letters rearranged gives 'DEEP SOUTH'.

'in' acts as a link.

Can you help me to learn more?

(Other definitions for deep south that I've seen before include "Region including Alabama, Georgia, Louisiana" , "part of US" , "part of America" .)

I've seen this clue in The Irish Times.

Want a hint initially instead of a full solution? Install my app