Thanks to a Volokh Conspiracy commenter, I discovered that in August 1960, the Democrat-controlled Senate passed a resolution, S.RES. 334, “Expressing the sense of the Senate that the president should not make recess appointments to the Supreme Court, except to prevent or end a breakdown in the administration of the Court’s business.” Each of President Eisenhower’s Supreme Court appointments had initially been a recess appointment who was later confirmed by the Senate, and the Democrats were apparently concerned that Ike would try to fill any last-minute vacancy that might arise with a recess appointment. Not surprisingly, the Republicans objected, insisting that the Court should have a full complement of Justices at all times. Of course, the partisan arguments will be exactly the opposite this time.
UPDATE: Updated with a link to the final vote on the resolution, 48 Democrats voting “yea”, 33 Republicans and 4 Democrats voting “nay.” Also, note that President Eisenhower had recess-appointed William Brennan to the Supreme Court in October 1956, just before the presidential election. With a winnable election coming up, Democrats obviously didn’t want a replay.