My advice is to take your time. You've seen a nice response to the initial protocol. Let it run for a couple of months, for at least one lab test cycle. A T level of 570 is good. Right now, during this ramp-up period, your free testosterone % is higher than normal. This will stabilize after a while and help lock in the number for T and Estrogen.
Again, let it run like this for as long as possible. It's easy to forget but there is nothing magical about any specific T level. There is no reason to believe you'll do better at, say, 800, then you will at 570. You now have more than you had to start AND your body is creating it "naturally" with all of the beneficial effects of stimulating the related metabolisms. For now, count yourself lucky: HCG Monotherapy would be THE preferred hormone therapy for men but it doesn't work consistently for many of us.
Regarding your question about best time of day to get tested, for me mornings are best, before breakfast. Before coffee even. The important thing is consistency. Choose a time and day, the same point in your injection cycle each time, for all lab work.