As another diarist has noted, when the new Senate convenes in January it will at first have a small majority of Democratic Senators, until the new Senator-elects have been sworn in.
That diarist suggested the opening minutes before the new Senators are sworn in could be used to push through the nomination of Garland.
But there are other equally creative options available, as described below...
Scenario 1:
The ongoing Senate could immediately adjourn for a week, before any new Senators are sworn in. This would require permission from the House: “Neither House, during the Session of Congress, shall, without the Consent of the other, adjourn for more than three days, nor to any other Place than that in which the two Houses shall be sitting.” The House would likely object to such a recess. But then Article II, section 3 kicks in(!): “[The President] may, on extraordinary Occasions, convene both Houses, or either of them, and in Case of Disagreement between them, with Respect to the Time of Adjournment, he may adjourn them to such Time as he shall think proper”. So Obama could adjourn them both for a week “to let tempers cool”. At which point, he could make recess appointments for ALL pending judicial vacancies, since the recess would be longer than three days.
Scenario 2:
Under Article I, section 5, clause 3, “Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members”.
So the continuing Senate could cite concerns about rigged elections and insist that no new Senator be seated until there was a verified hand-count of all the ballots. That maintains the Democratic majority for some time. If it turns out that states that make it hard to count votes that way can’t get their act together, “some” time could be a long time.
Would the Republicans scream bloody murder? Hell yes.
Would they do it if the shoe were on the other foot. I wouldn’t put it past them.
Would the precedent come back to bite us later? Probably — maybe someone else has thought this trough in more detail.
Food for thought...