Had Babe Ruth been a player in pretty much any decade other than the one he did retire in, his #3 would not have been reissued by the Yankees.
Derek Jeter’s #2 wasn’t officially retired until May 2017, but there’s no way the team ever would have even considered giving it to someone else. Paul O’Neill’s #21 isn’t retied, but the few times someone else wore it...it didn’t go well.
All of the other notable numbers were pretty much taken out of circulation after their most famous player ended their tenure with the Yankees. No one had #4, #5, or #7 after Lou Gehrig, Joe DiMaggio, and Mickey Mantle. However, seven other people wore #3 after Babe Ruth.
George Selkirk got it the season after Ruth’s final year as a Yankee. Though obviously not remotely on level of Ruth, Selkirk was a really good player in his career. He was a two-time All-Star.
Bud Matheny got the number starting in 1943 after Selkirk left. Matheny had a pretty solid rookie year, putting up a 112 OPS+. The following two seasons were a bit below average, and he was out of the majors after getting just three at-bats in 1946. Roy Weatherly got the number for two games in May ‘46, which was the extent of his Yankee career.
Later on in 1946, the number was given to Eddie Bockman. Bockman wasn’t the final player to wear #3, but that is where the shark was seemingly jumped.
Bockman had seemingly put up decent numbers in the minors, though Baseball Reference doesn’t have all of them. (Minor league numbers from 1939-42 can’t exactly be the easiest things to get a hold of.) He missed the 1943-45 seasons while serving in the military, but when he got back he managed to catch on with the Yankees in September 1946.
He made his Yankee and major league debut on September 11, 1946, wearing Babe Ruth’s famed #3. Bockman came in as a late inning replacement at third base for Billy Johnson. He went 0-for-2.
Bockman did not make another appearance until 12 days later. He walked and doubled in his first two plate appearances. Other than one additional error-assisted instance, those would be the only two times he reached base as a Yankee.
The following two days, Bockman went a combined 0-for-7 in a pair of close losses to the Red Sox. He finished his Yankee career 1-for-12 with one walk. That is good for a .083/.154/.167 line. He had a -11 OPS+ and a -15 wRC+.
Players have bad short stints and get jettisoned after a handful of games all the time. However, not all of them are seemingly random call ups wearing Babe Ruth’s number. Imagine last year if the Yankees hadn’t retired Jeter’s number, and instead gave it to Pete Kozma and his .311 OPS.
After the season, Bockman was traded to the Indians for a player to be named later. He played another 195 games across three seasons for Cleveland and the Pirates from 1947-49. He hit a little better than his .321 OPS with the Yankees, but not well enough to stick for too long in the majors.
Four more players wore #3 after Bockman. Cliff Mapes was the final one, before switching during the season when the Yankees finally retired it in 1948. Mapes actually went to #7, and was one of the final people to don that number before Mantle got it.
In some respects, it’s a good thing that the Yankees aren’t giving out numbers that are obviously about to be retired. On the other hand, we’ll never get the comedy of Brent Lillibridge or someone random wearing #6.