It would save time, and barely add any clutter (just three characters), to show the day of the week for all dates in tooltips such as the black-background tooltip here:
So, instead of showing:
Mar 10 11:00am
Tue Mar 10 11:00am
Also, in some cases, including the grey-outlined date display here, I’d argue that the same day of the week should be added there too to add clarity. This is a canonical, detailed display for information like this, and I don’t feel three characters is too much clutter added to provide useful info.
Often, I find myself clicking on the date in order to display the calendar so I can determine the day of the week for future dates, or mentally trying to calculate them, and it would be nicer to have it provided, at least in the tooltip.
for similar reasons, I think you should not hide the month and day. So instead of the tooltip shown, use this instead:
Wed Mar 4 11:00am