As to your inquiry about the best times to reach out to busy providers and what the best point of contact is initially, everybody is different so I can't really say anything outside of what works best for me.
I am fairly popular, I suppose, and get more appointment requests than I can actually accommodate: to be more specific, I only see 1 (at most 2) clients per day, and usually see friends for extended multi-hour appointments. I receive about double the numbers of requests than I can actually take on.
For me, email is always the best first means of communication and advanced booking is greatly appreciated. It is also best for you because I can almost always make an advanced request work and same-day appointments are not nearly as easy to fit in.
However, if you want to try your luck at same day, call or text me. And in terms of hours, just use common sense: calling a provider at 7am is probably not going to receive a response, and providers tend to wake later than the "normal" professional community. I don't answer my phone number 9am or 10am, but I do check messages, so if you call, just leave a voicemail or a text and I will get back to you.
Also, the more info you can leave in your initial contact, the better. Let me know when you want to meet, where you would like to see me, and what the best way to reach you is. Also, if you can include screening info, that's even better -- especially if you are reaching out on short notice.
Hope this helps!