WASHINGTON — Eleven years ago, when the Phillies chucked $50 million at free agent Jonathan Papelbon, then-general manager Ruben Amaro Jr. theorized that a contending team, especially one with payroll flexibility, must do all that it takes to protect late-inning leads. Amaro spared no expense, then, to try to spare the Phillies the psychological toll of not closing out victories.

“I think it’s essential that we win the games we’re supposed to win because of the commitment we’ve made to our starting pitching and just our pitching in general,” he said at the time. “It’s important we make sure we win the game when we have a late lead.”

It was true then, and it’s true now. Imagine how much differently the last few seasons would have gone if the Phillies didn’t have the fourth-highest bullpen ERA (4.76) or the third-highest blown saves total (72) in baseball since 2019. Or if they weren’t outscored 722-666 after the sixth inning and 189-162 in the ninth inning alone. Surely they would have made the playoffs a time or two instead of being 0-for-3 in the Bryce Harper era.

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To be fair, the Phillies have not neglected the bullpen. Under former GM Matt Klentak, they targeted relievers with experience. In the last couple of years, they prioritized hard throwers. But no matter the strategy, they didn’t hit on trades (Brandon Workman, Heath Hembree, David Phelps, José Alvarado, Sam Coonrod, Ian Kennedy) or get much return on free-agent investments (Tommy Hunter, David Robertson, Archie Bradley, and this year, Corey Knebel and Jeurys Familia).

Knebel blew his fourth save in 41 days — a span of 16 appearances — Tuesday night against the Miami Marlins, and a day later, interim manager Rob Thomson displaced him as the closer. For now, Seranthony Domínguez and Brad Hand will get the majority of the save opportunities. If neither wins the job outright, Knebel likely will get another chance.

But after 63 games, it’s clear that the Phillies are short at least one late-inning reliever. Even if Knebel rediscovers his curveball and harnesses his fastball command, and if Familia finds his power sinker, and if erratic lefty Alvarado turns around his season, and if Domínguez continues to dominate despite missing most of the last 2½ seasons, there still won’t be a comfortable level of certainty at the end of games.

The Phillies averaged 7.1 runs per game during their nine-game winning streak and built leads that were largely bullpen-proof. That isn’t sustainable over the final 99 games. And feeling a constant need to outrace the bullpen is stressful on hitters, to say nothing of a manager and a pitching coach.

It’s almost a lock, then, that president of baseball operations Dave Dombrowski will span the globe for bullpen help before the Aug. 2 trade deadline. The Phillies scouted veteran right-hander Trevor Rosenthal’s showcase in Miami last week. Their search will only intensify over the next six weeks, as they join other bullpen-needy teams in trying to make a deal for this year’s Clay Holmes.

Holmes has been the best reliever in baseball since the New York Yankees stole him from the Pittsburgh Pirates at last year’s deadline. There won’t be any stealing Pirates closer David Bednar, who would be the reliever prize this time around if Pittsburgh even elects to move him. The asking price will be steep, as it should be, for a 27-year-old right-hander with a 1.14 ERA, 11 saves in 12 chances, a $715,000 salary, and four more years of team control.

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Unless the Phillies reverse course on untouchable pitchers Mick Abel and Andrew Painter, they likely lack the prospect capital to land Bednar.

Several other closers on noncontending teams will be similarly valued because of their contract status. Baltimore’s Jorge López, Kansas City’s Scott Barlow, Detroit’s Gregory Soto, and Texas’ Joe Barlow are all controllable for at least the next two years. If their teams can’t fetch more than a midlevel prospect, they may choose to keep them.

It will be a few more weeks before the trade market truly takes shape. The addition of one extra playoff team in each league could create more of a seller’s market than in years past. Or maybe a few fringe contenders fall out of the wild-card race and decide to cash in a few chips. Market conditions are still fluid.

Regardless, here are three relievers who are worth watching as the deadline creeps up:

Mychal Givens, Cubs

The Phillies showed interest in Givens in November, before a 99-day lockout froze transactions and shut down the sport. Once the lockout ended, they pivoted to Familia and Hand, while Givens signed with the Cubs for $5 million.

Givens’ numbers over the years have mirrored former Phillies reliever Héctor Neris. Givens, 32, is 11 months younger than Neris and made his major-league debut 10 months after Neris. Both are right-handed. Both throw cutters, albeit from different angles. Givens has a 3.47 ERA and 497 strikeouts in 419⅔ innings in his career; Neris has a 3.44 ERA and 549 strikeouts in 434 innings.

It may not be the solution to the ninth inning, but Givens would be another late-inning reliever with a track record.

Michael Fulmer, Tigers

The Tigers won four consecutive AL Central titles from 2011 to 2014. But by the All-Star break in 2015, they were a .500 team. It marked the last time that Dombrowski behaved like a seller at the trade deadline.

Over the span of a few days, Dombrowski traded David Price, reliever Joakim Soria, and Yoenis Cespedes. Fulmer, a former first-round pick of the New York Mets, was among the players Dombrowski got back in the deal for Cespedes. A year later, after Dombrowski got fired and joined the Boston Red Sox, Fulmer was the Tigers’ best pitcher and the AL Rookie of the Year.

Since then, Fulmer had Tommy John elbow surgery and reinvented himself as a reliever. Over the last two years, he has a 2.82 ERA, 15 saves, and 97 strikeouts in 92⅔ innings. He’s setting up for Soto, but with free agency looming for Fulmer after the season, the last-place Tigers almost certainly will get what they can for him.

David Robertson, Cubs

When the Phillies signed Robertson to a two-year, $23 million contract in the “Stupid Money” offseason of 2018-19, it looked like a safe move. The veteran right-hander was among the most durable relievers in baseball, making at least 60 appearances in nine consecutive seasons.

It was little more than rotten luck that Robertson tore a ligament in his elbow after only seven appearances for the Phillies in 2019 and needed Tommy John surgery.

At age 37, Robertson is back and as good as ever. He signed a one-year, $3.5 million deal with the Cubs, won the closer job, and had a 1.59 ERA and seven saves in nine chances through Wednesday.

Robertson is one of a few former Phillies relievers who likely will be on the move before the deadline. But unlike Texas Rangers lefty Matt Moore or Kennedy, who is with the Arizona Diamondbacks, a reunion seems plausible with Robertson because his Phillies tenure was marked by injury, not ineffectiveness. He also was signed by Klentak, not Dombrowski.