How precisely the "clay pigeon" might be launched: Via Pajamas Media, a memo from former Secretary for the Majority and Minority in the U.S. Senate Elizabeth B. Letchworth (emphasis added):

If 60 Senators vote to make S. 1639 the pending business in the Senate on Tuesday, June 26 [As they did this afternoon--NA], Sen. Reid will immediately take the floor of the Senate and offer a 1st degree amdt. to S. 1639, and ask for the yeas and nays. This guarantees a roll call vote will occur on this amdt.
He will then send a 2nd degree amdt to the desk and ask for the yeas and nays. This achieves the same for the 2nd degree amdt. He will then ask that the amdt. to be divided.
This amdt will automatically be divided into sections that can stand alone as individual amdts. In other words, his second degree amdt. may have 18 sections [Though I've heard that there are now maybe 24 amendments in this package--NA], and by using this procedure, he has now put in place 18 amdts. Since the yeas and nays were ordered, so he has now been guaranteed a roll call vote will occur on all of these sections by using the Rules of the Senate.
These sections are presumably going to contain the text of amdts he believes Republican and Democratic Senators wanted to offer when the Senate debated the last immigration bill a few weeks ago.
Now no other amdts are in order since he has placed his amdts in these specific places in the amendment process allowed by the Senate Rules. No other Senator can stop him from using these maneuvers since Sen. Reid has priority recognition over all other United States Senators. By Sen. Reid using this amendment process and calling for a division of his amdt., he can guarantee Senators that votes will occur on issues he has chosen to offer and on issues he believes will please enough Senators as to earn the 60 votes necessary to invoke cloture and pass the bill.
Sen. Reid will then send a cloture motion to the desk setting up the cloture vote for Thursday. The bill is now pending but no Senator can offer amdts because he is blocking them by using his priority recognition and offering his own amdts.
If 60 Senators vote on Thursday to limit debate on S. 1639, at the end of the 30 hours allowed by Senate Rule 22, votes will automatically begin, without further debate, on all of the Reid amdts in a back-to-back sequence. The final vote will be a vote on Final Passage of S. 1639, the Kennedy Immigration bill.

This move may be made at 3:50...