I don't know who their best players are, but you can bet they are planning to have what they think is to their best political advantage on at prime time, and in the morning before the daily programs cut away.
They have tomorrow to work with as well, and almost certainly Friday morning. On Friday morning, Frist may telegraph his plan for asking to vote on the nominee. He won't ask to take the vote then, just indicate that at some point, there will be a request for unanimous consent to move to the vote. He may say that "next week, we will be asking, on a periodic basis, for unanimous consent to go to the vote. We aim to have had a full debate, blah blah blah"
There is no rule that cloture MUST be advanced. Most bills are voted on without a cloture motion. Cloture is a way to get to the vote WITHOUT unanimous consent.
The timing of asking for the vote is a fluid matter. I don't know for sure how Frist will gauge the timing - opinion polls? news broadcasts? pundit suggestions? editorials? or just his sense that the public has seen enough debate, and the public will "accept" or "understand" putting a question to the chair. Maybe it'll take until next week Friday. We do know that Frist has put no time certain to the matter. The nominee will be fully debated.
Hillary always shows up for Prime-time but I think Scumee has already given her speech.