Joe Harris is one of those incredibly useful role players every NBA team would be lucky to have. He was also unemployed not too long ago.

Jan. 12, 2016, was a miserable day for Harris. It started with foot surgery. That was the best part of his bad morning. He was still in recovery when he got a call from the Cleveland Cavaliers and heard one of the most dispiriting phrases in the English language: You’ve been traded to the Orlando Magic. But he never made it to Orlando. He was waived first. Harris came to the hospital on one team and left after being cut by another team.

He wouldn’t have a job for the next six months.

When the Cavaliers played the Golden State Warriors in the Finals, he was playing in pickup games at the New York Athletic Club. The classic Game 7 that delivered Cleveland its first championship? He watched from his childhood home in rural Washington. Harris put his odds of returning to the NBA at 50/50.

“I was on the outside looking in,” he said.

But he could see the sport changing in front of his own eyes, and it wouldn’t be long before every team would need someone who could stretch the court by shooting 3-pointers—someone like Joe Harris. He tried to be optimistic. “At the same time, you have to be realistic,” Harris said. “There aren’t a lot of second chances for second-round picks.”

His time on the NBA unemployment line ended with a tentative job offer from the Brooklyn Nets, and he played so well after signing for the minimum salary that the Nets offered him another, more lucrative deal last summer. In a span of two years, Harris earned himself a 750% raise.

There are few NBA players whose values have increased more and even fewer whose careers explain the remarkable ongoing evolution of the league as neatly as his. This bearded guy in Brooklyn named Joe who goes unnoticed on the subway also happens to be a bellwether of the sport.

“In the modern NBA,” said Nets coach Kenny Atkinson, “he personifies what you’re looking for in a wing.”

It’s worth paying attention to someone like Joe Harris because of what he reveals about the rest of the sport. Star players transcend the game. Role players reflect it.

Not just a shooter anymore, Joe Harris has become an elite finisher, which in turn makes him a better shooter. Photo: kyle terada/Reuters

The style of play that has infected the NBA in this pace-and-space era demands a certain style of player. Harris signed a two-year, $16 million deal over the summer because he’s turned himself into such a player.

There’s a job in the NBA for anyone who can make two of five 3-pointers and one of two open 3-pointers, and Harris is making 44% of his threes and 53% of his open threes. But he wouldn’t be nearly the player he is today if he were only a good shooter. The people closest to Harris describe him as a player with a relentless drive who changed his game to take advantage of the ways the game was changing.

“I think people all around the league are scratching their heads as he keeps getting better,” said Mark Bartelstein, his agent. “Everyone was like, OK, he made it, but he probably won’t be a rotational player. But then he became a rotational player. Then he became a really, really good rotational player. Then he became a starter. Now he’s one of the best shooters in the league.”

There are few shooters in the NBA who take and make as many catch-and-shoot 3-pointers as Joe Harris. Photo: Kathleen Malone-Van Dyke/Associated Press

The son of a high-school basketball coach, the 6-foot-6 guard was a four-year college player at Virginia drafted with the 33rd pick in 2014 by the rebuilding Cavaliers. He might have gotten a chance to play right away if they were as bad as they planned to be. But not long after the draft, there was a slight change in Cleveland’s strategy. “That was before LeBron,” said Virginia coach Tony Bennett.

There wasn’t much use for a rookie on a title contender. For the first time in his life, Harris was glued to the bench. He worked behind the scenes on the little things that could make him better—former Cavs guard Mike Miller gave him lessons in getting his shot off milliseconds faster—but he wouldn’t get a chance to showcase those skills for the team that drafted him after his day from NBA hell.

He moved to New York and lived with a friend of a friend on the Upper East Side of Manhattan. He went through his injury rehab not knowing where in the world it would lead him.

It turned out to be across the bridge.

The Nets were a desperate team in search of desperate players. After a disastrous 2013 trade robbed their stash of draft picks, their only means of improvement was the scrapheap. In the summer of 2016, when a one-time explosion of the salary cap created an unprecedented spending frenzy, they signed Harris to a minimum contract worth less than $1 million in his first season. It was the least amount of money they could legally pay him.

Atkinson only realized once he got to know Harris that he’d been deeply misvalued somewhere along the way.

“This is not some miracle of some guy who couldn’t play,” Atkinson said. “He could play. But he got an opportunity here.”

The Nets wanted Joe Harris (left) to emulate Kyle Korver (right). Photo: ken blaze/Reuters

The Nets asked Harris to emulate Kyle Korver. But there was one big difference between them: Korver is 37 and Harris is 27.

He’s entering the prime of his career at the exact moment his profession was putting a premium on players of their size and skill. By spacing the court and stressing the defense, brilliant shooters tilt the balance of power on every possession toward the offense. But the job is trickier than it sounds.

It required Harris to make a higher percentage of his catch-and-shoot 3-pointers (he’s improved from 39.2% to 41.2% to 46.4%), to work harder to get shots (he covers more ground on offense than any Nets player) and to expand his game with defenses running him off the 3-point line (he’s an elite finisher who now shoots 65% around the rim).

Harris slowly became a player who can plug into any team. That became clear this summer, when he was a free agent, and he learned his services were in high demand for someone who couldn’t get a job two years ago.

He left money on the table to re-sign with Brooklyn. He’s only gotten better with the Nets this season. And he still walks to the arena from his apartment.

Write to Ben Cohen at ben.cohen@wsj.com