Corker DOES NOT SUPPORT THE BILL. His vote was purely procedural. In order for the minority party to retain their right to filibuster, at least ONE member of the minority party MUST vote for the majority’s bill.
That's false. If you are determined that it is true, I defy you to find and cite authority for the proposition. Here are links where you would find the rule, if it existed: Link to Senate Rules and Link to Riddicks Senate Procedure.
Sometimes (and this is one of those times) a procedural vote is tantamount to a substantive vote. If this amendment failed to obtain cloture, then it would be stuck in infinite debate mode. If Corker wanted debate to continue, he would not have voted to limit debate with voting on a budget point of order or passage of the amendment to start around noon Tuesday.
The power to move to reconsider a vote does attach to those on the prevailing side of the vote, and there have been many motions to reconsider FAILED cloture motions (sort of stupid to do so, since cloture motions are free anyway (i.e., can have a hundred of them on the same subject, f Senate leadership wants to), a do-over is the same as have a fresh vote), but there is no sense whatsoever to having a revote on a motion to limit debate. Senate Rules XXII itself describes how debate can be extended beyond the 30 hours post cloture. But in The instant case, the Senate has unanimous agreement that post cloture time is up, notwithstanding the clock, at noon tomorrow.