Technically it wouldn't matter, but of course by posting in a fresh Open Thread you get better visibility.

(So sometimes it is strategic to wait even if the old Open Thread is still "valid", but already full of comments.)

And you are right about the time zones; although there is no hard rule when exactly the threads appear, "one day later" is a good rule of thumb.

It's not automated, so the new one will be posted whenever someone gets around to it, which they normally do pretty quickly. You'll get more people looking at your post if you put it at the beginning of a new open thread.