Daily customer to begin with, one with a list? (7)

I believe the answer is:

cleaner

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'daily' is the definition.
(dated term for a woman cleaner)

'customer to begin with one with a list?' is the wordplay.
'to begin with' says to take the initial letters.
'one with' indicates putting letters inside.
'a' becomes 'an'.
'list?' becomes 'leer' (I can't explain this - if you can you should believe this answer much more).
The first letter of 'customer' is 'c'.
'an' inserted inside 'leer' is 'leaner'.
'c'+'leaner'='CLEANER'

Can you help me to learn more?

(Other definitions for cleaner that I've seen before include "Char; less dirty" , "Charlady" , "More hygienic" , "less sleazy?" , "Worker" .)

I've seen this clue in The Times.

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