There’s a precedent for doing it for cert applications, but I’ve not heard of doing it for cases that are actually under appeal.
After Abe Fortas resigned from the Court, there were a number of pending cert applications that had votes from 3 justices, but had been voted against by the other 5 justices. It takes 4 votes to get cert, so there was a possibility that the new justice would vote in favour of cert. The Court put all those 3-5 cert applications into a file, “Hold for Justice X.”
It took a while before Justice X appeared. Fortas resigned on May 14, 1969. Nixon then had trouble getting his nominations approved to replace Fortas: the Senate rejected his first nominee, Clement Haynsworth by a 55-45 vote, and then rejected Nixon’s second nominee, G. Harrold Carswell, by a 51-45 vote.
Finally, on May 12, 1970, the Senate confirmed Nixon’s third nominee, Harry Blackmun, by a 94-0 vote.
So the Court was short a member for almost exactly a year. The “Hold for Justice X” file must have been getting big.
(Source: The Brethren by Woodward and Armstrong.)
As an aside, just to show how much things have changed in Washington, look at the breakdown of the two votes to reject:
Haynsworth: Of the 55 Senators who voted against him, 38 were Democrats and 17 were Republicans; of the 45 who voted for him, 19 were Democrats and 26 were Republicans.
Carswell: Of the 51 Senators who voted against him, 38 were Democrats and 13 were Republicans. Of the 45 who voted for him, 17 were Democrats and 28 were Republicans.