What time of the day...? assumes that the listener knows which day is in question: '[You've already told me that he left last Monday, but]...'
What time of day...? just asks when, or asks for a broad answer - 'In the morning/afternoon/evening' would be appropriate responses.
If you're just asking when, it might be preferable to say 'When was it that the man woke up?' or just 'When did the man wake up? But the wordier options are quite grammatical.
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