Mad dog had a trilby, for the midday sun? (5,8)

I believe the answer is:

broad daylight

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

'midday sun?' is the definition.
'broad daylight' can be an answer for 'sun?' (I have seen 'Full sun ' mean 'broad daylight' so perhaps 'sun' could also mean 'broad daylight'). I am unsure of the 'midday' bit.

'mad dog had a trilby' is the wordplay.
'mad' indicates an anagram.
'dog'+'had'+'a'+'trilby'='doghadatrilby'
'doghadatrilby' is an anagram of 'BROAD DAYLIGHT'.

'for the' is the link.

Can you help me to learn more?

(Another definition for broad daylight that I've seen is " Full sun".)

I've seen this clue in The Independent.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.