Brian Mazone's near-miss effort to make it into a major league game was chronicled by The Post this summer. (Sandy Huffaker for The Washington Post)

On May 23, in need of an additional bullpen arm, the New York Yankees called up reliever Ryan Bollinger from their Class AA team in Trenton, N.J. The news, delivered by his Trenton manager, left the 27-year-old lefty, a veteran of multiple independent leagues and pro ball in Australia and Germany, in tears. When Bollinger met the Yankees in Arlington, Tex., for the finale of a three-game series against the Rangers, the team gave him uniform No. 61, which he wore as he watched from the visitors’ bullpen, never appearing in a game the Yankees would lose, 12-10.

The next day he was sent back to Trenton.

And just like that, Bollinger was granted membership — provisional for now — in a strange, little-known and not-exactly-desirable club: that of the “phantom big leaguers,” a loosely defined group of star-crossed, largely obscure minor leaguers who, in one way or another, were promoted to the majors but never appeared in a big league game.

Nearly three weeks ago, The Post published a lengthy feature story on Brian Mazone, a minor league journeyman and soft-throwing lefty, who in 2006 was tabbed by the Philadelphia Phillies to make a spot start in a game against the Houston Astros, only to have the game rained out. Returned to Class AAA because the Phillies no longer needed him, Mazone was never added to the Phillies’ roster and would never make it back to the majors — and thus does not appear in any official Major League Baseball register. In other words, his one and only shot at the big leagues got rained out.

While Mazone is likely the only player in history to have suffered that precise set of circumstances, he is far from alone in one larger sense: Baseball history is riddled with examples of players who made it to the cusp of the majors, or even had their promotions made official with a roster move, but never made it on the field. And in this age of 10-day disabled lists, constant bullpen turnover and frequent roster moves, it seems to happen several times a year.

In the wake of the Mazone story, The Post heard from dozens of readers and tweeters offering the names of players who had met similar fates. And one of those readers, Bill Hickman of Rockville, Md., happens to be a longtime member of the Society for American Baseball Research (SABR) who maintains several databases of what he calls “Near Major Leaguers” — the entries, when taken together, number in the thousands.

Two of Hickman’s databases — featuring players who appeared on the spring training rosters of big league clubs, and nonroster invitees in spring training camps between 1950 and 1999, none of whom appeared in a regular season MLB game — contain the names of more than 6,500 players.

But it is the third database — featuring 429 players (for now) who have appeared on a major league roster during the regular season but never appeared in a game — that hammers home just how many minor leaguers creep right up to the edge of big league status but for one reason or another never make it into a game and thus get their names in the record books.

“I admire the fact they were able to get that close,” Hickman said when asked what drew him to those near miss players. “I really appreciate the fact there are so many young men who go into the minor leagues, and so few who even get that far.”

As you would expect, some of the stories behind the names on Hickman’s list are excruciating in their near miss details. In 1979, catcher Harry Saferight was a September call-up of the Pittsburgh Pirates, and though he never got in a game, three different times he was in the on-deck circle when the game ended. Outfielder-catcher Lee Robinson spent more than a month with the 1974 Los Angeles Dodgers without getting into a game. In 2011, catcher Brian Jeroloman also spent more than a month with the Toronto Blue Jays, but didn’t get into a game — because he had a broken hand. (The Blue Jays never bothered to place him on the disabled list because with expanded September rosters, they didn’t need his roster spot.)

“I always tell people not to feel bad for me,” Jeroloman, who kept playing until 2016 but never made it back to the majors, once told MLB.com. “This is part of a baseball. It’s part of the life I chose.”

Hickman is among those who sometimes refer to these near-miss players as the “Bill Sharman Society” — a designation coined by longtime sportscaster, political pundit and SABR member Keith Olbermann — in honor of the Naismith Memorial Basketball Hall of Famer who won four NBA titles as a guard for the Boston Celtics and another as the head coach of the Los Angeles Lakers.

Sharman, who died in 2013, was also an accomplished baseball player, and in September 1951, as a first baseman-outfielder in his second year in pro ball, he was called up to the majors by the Brooklyn Dodgers, on whose roster he spent nearly two weeks — culminating with the Oct. 3, 1951, playoff game against the New York Giants that was made famous by Bobby Thomson’s pennant-winning “Shot Heard ‘Round the World” — but never got into a game.

He continued playing baseball until 1955, by which time he was already a two-time NBA all-star, but never made it back to the majors. This makes Sharman the most famous member of the Phantom Big Leaguers Club, or if you prefer, the Bill Sharman Society.

Olbermann said he became interested in phantom big leaguers while researching Irving Lewis, a dead-ball-era catcher who spent time on the Boston Braves’ roster in 1912 but never appeared in a game, and whose 1912 baseball card is considered one of the scarcest in the world. Lewis’s story resonated for him, he said, because as a teenage New York Yankees fan, Olbermann had seen pitching prospect Ed Ricks called to the majors in 1977 only to be sent back without appearing in a game, never to make it back up again.

Ricks’s fate, Olbermann said, “seemed elementally, fundamentally unfair.” When he realized how many players had met similar fates over the years, it resonated with the lifelong baseball fan in him. “This wish, come true: ‘If I could only be on the team, just once. Wouldn’t even have to play!’ Every kid ever interested in sports has had that thought,” Olbermann said. “It is viscerally understood by all of us.”

But even Hickman’s exhaustive research has gaps. Because it includes only players who never appeared in a game (thus never gaining entry into the official record-keeping), it doesn’t include, for example, Larry Yount, one particularly snake-bitten pitcher whose story was referenced within The Post’s Mazone profile.

In 1971, Yount, the brother of eventual Hall of Famer Robin Yount, was set to make his big league debut for the Houston Astros, but hurt his elbow warming up in the bullpen. Called into the game, he jogged to the mound, took a few more warmup tosses — but couldn’t continue. He walked off the mound and into the dugout, never to be seen in the majors again. But because he had been announced as the Astros’ new pitcher, Yount’s entry into the game became official, and in the record books he is credited with one pitching appearance but zero batters faced.

Of course, the SABR database of “Near Major Leaguers” also misses Archibald Wright “Moonlight” Graham, who played a single inning in right field for the 1905 New York Giants but never got to bat — and whose story was memorialized in the book “Shoeless Joe” and the movie “Field of Dreams.” Players like Graham, whose big league careers consist of just one game (or, as that phenomenon is known, a “cup of coffee”), have their own researchers, databases and mystique.

As for Ryan Bollinger — the 27-year-old Yankees farmhand whose first time on a big league roster, this May, ended without any game action — there is still time for him to get his name stricken from the ranks of the Bill Sharman Society. He went a combined 8-6 with a 3.87 ERA between Class AA and Class AAA this season. Perhaps he is a future “cup of coffee” candidate, or perhaps he goes on to a lengthy, fruitful big league career.

But Bollinger already has two strikes on him. On July 31, more than two months after his first call-up, Bollinger was promoted to the majors by the Yankees for a second time in 2018 — and again, he was sent back down the next day without having appeared in a game.