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can i say something like "have a good rest of the day!" during the day? does it mean to have some rest now that it's getting late or to spend a nice time during the "rest part" of the day?
Comments 
Hi,
can i say something like "have a good rest of the day!" during the day? You can, but it's very casual and informal English.

Less casual is eg 'Enjoy the rest of the day!'

does it mean to have some rest now that it's getting late No, definitely not.

or to spend a nice time during the "rest part" of the day? Have a nice time during the remainder of the day.

Clive
Well said clive
Kudos Clive!!