ELIZABETHTOWN, Ky. — One Hall of Fame career.

One model of bat.

One family’s baseball legacy.

Leslie Wayne Pinkham came so close to playing in the majors in the 1950s, but his dream was never realized because of a freak injury to his thumb.

The career of his son, Leslie William Pinkham, also came to an early end in the 1970s in the Reds organization because of a shoulder injury.

Father and son were both catchers.

The Pinkhams’ legacy, though, lives on in a Hall of Fame way with the Louisville Slugger P72 model bat, designed to Leslie Wayne’s specs in 1954.

For every one of Derek Jeter’s 3,316 hits, he has used the P72 — a rarity in baseball because players like to experiment with different bats. Jeter is 10th all-time in hits. In his 19-year career, that 11,968 trips to home plate — all with the same model of bat.

Throw in another 734 postseason plate appearances, 200 hits and you have an amazing body of work with the P72, a baseball marriage that is unheard of these days.

“The reason I use that bat,’’ Jeter told The Post recently, “is because when I signed, it was shaped like my aluminum bat that I used in high school, and my entire career I’ve never swung another bat. It just felt right.’’

Jeter offered a smile and said, “So far, it’s been all right, but there’s always room for improvement.’’

Many major leaguers use the P72. The elder Pinkham and his son never made the major leagues, but his bat is there every night. Hall of Famer Cal Ripken often used the P72, saying he liked the combination of “balance and barrel.’’

That’s a Cooperstown ton of hits from one model of bat — one with a thin handle and a medium-sized barrel that offers balanced weight distribution.

“I can’t think of any other ballplayer who has just swung one particular model of bat his entire career like Derek Jeter has,’’ said P.J. Shelley, the tour and programming director for Louisville Slugger. “We take a lot of pride in the fact that Jeter, who is in the top 10 in hits all-time, has so much appreciation for the product.’’

“It just has such a great feel to it,’’ Jeter said. “I always thought the ‘P’ stood for Tony Perez.’’

No, Derek, the P stands for “Pinkham.’’

***

This is a story with many twists and turns, like the winding mountain roads near Elizabethtown, about a 40-minute drive from Louisville.

The Pinkham family, you see, has one more turn at-bat.

Bill’s son, Leslie’s grandson, Zeke, 16, is a talented catcher who has already committed to the University of Louisville as a high school junior.

The original Pinkham bat was designed and created on May 24, 1954, for Leslie Wayne Pinkham. Les played five years in the minors with the Tigers and Cardinals, hitting .259 with 37 home runs.

There is no record of Leslie William Pinkham getting into a major league game, but he was on the verge of making the majors, and family members say he even was a September call-up. His career came to an end when he was injured while working an offseason job. His thumb was severely cut by a pane of glass, causing permanent damage.

From that minor league stint, though, this Hall of Fame bat was born.

After baseball, Leslie Wayne became a successful businessman, starting a car dealership in Elizabethtown. He died in 2009.

His son Bill was drafted in 1970 by the Reds in the fourth round, out of nearby Jefferson, Ind. At 6-foot-1, 195 pounds, Bill was a strong, fearless man. He had an incredible high school career, and the Reds were building a dynasty, the Big Red Machine.

His shoulder injury came in a minor league game when he was briefly put in the outfield. His baseball dream ended in 1972. The Reds survived. They had a catcher by the name of Johnny Bench.

This past Dec. 26, Bill died after a battle with cancer. He was 60 years old.

Now it’s up to Zeke to carry on the family tradition.

“Zeke was born to play baseball,’’ his mother Alicia said as the family poured through Bill’s detailed scrapbook in their living room, with the MLB Network on in the background.

Zeke knows all about his grandfather’s bat and Jeter.

“My dad and Papa Les told me all about it,’’ Zeke said with an easy smile. “He was proud that Derek Jeter used his bat. My dad told me that Mickey Mantle used the bat, too.’’

Imagine that. Jeter and the Mick bound to a baseball family in Hardin County, Ky.

***

It is in Louisville where all of Jeter’s bats are made at the Louisville Slugger Museum & Factory on 800 West Main Street, home of the Hillerich & Bradsby Company.

In 1884, in his father’s woodworking shop, 17-year-old Bud Hillerich first turned a bat for Pete “The Louisville Slugger’’ Browning after going to a game and seeing Browning break his bat. Browning got three hits his first day with his new bat.

Word spread quickly among the players, and the company has been making baseball bats for the pros ever since.

Danny Luckett has worked for Louisville Slugger for 44 years, one of four CNC (computer numerical control) operators in the Pro Bat Department. Those operators run the computerized lathe that produces all the Louisville Sluggers for major leaguers. Until the 1980s, bats were hand-turned for pro players. Luckett is the last man at the company who made bats that way for the pros.

“It used to take 15 minutes to make a bat by hand,’’ Luckett said. “With this machine, we can make 12 bats in that time.’’

Luckett, manning the machine, can turn a 3-inch round, 37-inch long billet of northern white ash or hard maple into a major league bat made to the most specific proportions. The majority of bats Jeter has used over his career have been made by Luckett.

About 12 years ago Jeter came to visit the factory with Yankees first baseman Tino Martinez.

“They had a great time,’’ Luckett recalled. “I think what they liked the best was the fact they could go to the mall and nobody had a clue as to who they were.’’

Luckett started working for Louisville Slugger just about the time Bill Pinkham was drafted by the Reds.

The original P72, the one that was hand-turned on a lathe and built to Les Pinkham’s exact specs in 1954, now sits in the bat vault of the museum with thousands of other original bat models.

“This is the Fort Knox of Louisville Slugger,’’ Shelley said.

The Fort Knox sits between Louisville and Elizabethtown.

“Every single bat you see in front of us is unique in it’s own way,’’ Shelly said, “different knob, different handle, different barrel. These are all the original bats. You never want to make a copy from a copy.’’

I pick up the original P72. It feels comfortable and balanced in my hands. I immediately understand why Jeter chose this bat for his inside-out swing when he signed with the Yankees 21 years ago.

This is how Louisville Slugger catalogues its bats.

The process starts with a letter, followed by a number. The letter represents the last name of the ballplayer who created the specific model, and the number is the sequential order within that particular letter. Leslie Wayne Pinkham was the 72nd player with his last name starting with the letter P to have a bat made to his specifics.

“Here is the A99,’’ Shelley said. “That’s Hank Aaron’s model. He was the 99th player with the last name that began with A that created a model for us.’’

I’m soon picking up the M110, Mickey Mantle’s bat. The M in the M110 does not stand for Mantle, however. It stands for Eddie Malone, who played for the White Sox and came up with the design in 1944. Curtis Granderson uses that same model of bat today.

In 1927 Babe Ruth sent a broken bat back to Louisville Slugger so they could make an exact duplicate. It was a Lou Gehrig model. This wasn’t just any bat. The bat has 21 notches around the logo, each notch representing a home run hit during the season Ruth set a new major league record with 60 home runs. The bat is on display in the museum.

Mets’ star David Wright has visited the museum.

“We busted out Joe DiMaggio’s bat for him,’’ Shelley said. “He was beside himself.’’

***

Jeter has described his injury-plagued 2013 season as a nightmare. Yet, a baseball season offers many discoveries — like the tale of Leslie Wayne Pinkham’s bat and the promise ahead for young Zeke.

Bill Pinkham taught his son to play the game the right way.

“He was old school,’’ explained Zack Pinkham, Zeke’s brother, who is 24 and has taken on the role that his father did so well, being there for every one of Zeke’s games, even driving him to Georgia this summer for the prestigious Perfect Game-East Cobb Invitational, going so far as to sit in the exact same spot his dad would sit during games to be a comfort for his brother.

On the night I saw Zeke play in a fall league game, he homered, tripled and singled.

Zeke said he still can hear his dad in his ear, and when he looks over his shoulder he expects to see his father at the game. He then shows me a video on his phone where his dad is shouting encouragement to him, a keepsake from this modern age.

“I wish he was still here to get on me, to throw me BP,’’ Zeke said. “I struck out the other day and I could hear my dad cussing, I started laughing. My dad was just a tough, tough man, just like my Papa Les. I didn’t even know my dad was sick until he couldn’t walk. He came to all the games, he had cancer, it was 105 degrees outside, and he was coaching third base.’’

“He scheduled his chemo treatments around baseball,’’ Zack said.

“Baseball was first, then chemo,’’ Zeke said. “He missed his last chemo so I could go to a showcase at the University of Tennessee.’’

Zeke Pinkham has an inside-out swing like Jeter. He is looking to build upon the high school success he had last year when he batted a stunning .509 with 56 RBIs. His summer league team, the EvoShield Canes (16U) out of Virginia, won the Perfect Game World Series, finishing 48-3. Another strong junior season then a big year in his senior season of high school, and, who knows, anything is possible.

Zeke knows he has the baseball bloodlines, the ability — in the offseason he catches Blue Jays’ pitcher Steve Delabar, who lives nearby — and the mental fortitude needed to succeed. Now it is just a question of making it happen, fulfilling his family’s baseball dream that first began 60 years ago.

“I have the mindset that I am going to make it to the majors,’’ Zeke said. “That’s my dream.’’

Then he said these words of family honor.

“When I get my first major league at-bat, I’m going to the plate with a P72.’’

Papa Les and his dad would be proud.

Something tells me, Derek Jeter would like that, too.