The phrase 'the rest of the day' is correct and usable in written English. You could use this phrase when you want to describe the remainder of a current day or when referring to the future: I plan to spend the rest of the day reading and relaxing.
The rest of the day looked promising.
The rest of the day, stay away.
The rest of the day flew by.
The rest of the day passed quickly.
And the rest of the day?
The rest of the day was spent working on projects.
These I wore for the rest of the day.
Then it's Radio 4 for the rest of the day.
What happens for the rest of the day?
She'll take you through the rest of the day.
I spent the rest of the day at the cathedral.
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MA of Applied Linguistic, Maquarie University, Australia