Date picker showing by week, is showing the month necessary?

I am trying to design a date picker for booking an appointment, and I think that, the most natural way of choosing a date is not by a calendar (showing full month) but by showing by week. Currently my design looks like this:

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Which shows, weeks starting by monday to sunday, and a bar on top that says either this week, next week, past week, two weeks away from today, …

Problem is, should the user know the month he is in or is knowing the week enough? I understand that if the user is trying to book an appointment next week or the other, showing the month won’t be necessary, but what if he is trying to book an appointment eight weeks from today?

I thought of adding the month/months to the ‘week’ bar, like the picture below shows, or of giving the option of showing a calendar, but I think none of this would be the most intuitive user experience design. What should I do?

enter image description here

Thanks