I think it would be worth it to contact the lady you are interested in seeing, and find out if she would be able to do an early morning date.  With advanced noticed, she may be able and willing to accommodate you.

I have my hours posted as 10a-10p, mainly because I really prefer to only meet someone during that time and just dont feel like being out on the road late at night, so Id like the majority of my dates to be during that time. But if someone were to contact me, especially if they are visiting from out of town and its their only opportunity, then I would definitely be flexible and plan for being up early or out late.  So, its worth a shot to just contact ladies you are interested in and ask