There’s an official letter on the desk of the character Jeff Bridges plays in “Hell or High Water.” His name is Marcus Hamilton, he’s a Texas ranger, he’s looking at a mandatory-retirement notice.

A storied career – chasing down thieves and killers, fugitives and scam artists – and now he’s counting down his last days.

“He’s reaching the end of the line,” Bridges says. “And I think that’s something that a lot of folks can relate to – giving up what they’ve done all their lives.

“What are they going to do now?”

Not that Bridges, 66, is contemplating any kind of retirement himself.

“In my job, people work up to their death bed,” he says, on the phone, in a car, in Austin, Texas, recently, doing interviews to promote his crackerjack contemporary western. “But as you get older, certainly – I find myself thinking about my mortality more these days. I guess there are different roles that I’m age-appropriate to play now, that’s a part of it. But just generally, getting old, you start to think about it. Your body, your mind, time’s passing. I’ve got a couple of grandkids now.”

In “Hell or High Water,” now showing, Bridges’ white-hatted, wisecracking lawman and his partner (Gil Birmingham) are chasing down two brothers (Ben Foster, Chris Pine) on a wild bank-robbing spree in the dusty towns of west Texas. The movie, directed by David Mackenzie from a screenplay by Taylor Sheridan (he played David Hale on “Sons of Anarchy,” he wrote “Sicario”), is easily one of the best of the summer.

There are car chases and shootouts, to be sure, but for once, they make sense – the action fits the context of a bigger, human story, and the characters, all of them, are wholly formed, flawed, real.

“I’m certainly pleased with this one,” Bridges says. “I saw it last night again. … It put a smile on my face.”

There’s a scene in “Hell or High Water” that may be one of the best things the six-time Oscar-nominated, one-time winner (for his boozing itinerant country singer in “Crazy Heart”) has ever done. It comes near the end. Marcus is clambering up a hill, closing in on one of the brothers – to say much more would be saying too much.

“I don’t know about you, but I like to see movies for the first time knowing as little as possible,” Bridges says. “At the same time, my task here, right now, I’m like the barker in front of the carousel – trying to get people to come and see a show.

“But I want them to have the best experience that they can, which means knowing as little as they possibly can about it. So maybe I can talk about that moment in more general terms?”

OK, we’ll abide.

“There are a lot of different ways to approach acting, and one way is to just kind of turn yourself over to receive ideas,” Bridges says. “You could be in your hotel room, learning the lines and imagining what the scene is going to be like and all this stuff, but once you get out there, it’s quite a bit different than you had imagined. And you can sometimes struggle with that. In the script, it might be a sunny day, and then when you get out there it’s raining, and you can waste a lot of energy bitching about the rain, or you can let that inform your work. Maybe do a rain dance.

“And scenes that involve emotion – you might imagine what it’s going to be like, but the task at the moment when the cameras are rolling, you just let that go, and let the emotions have their way with you. And you never know what’s going to happen.”

Whatever happened, Bridges’ emotional moment in “Hell or High Water” is genius.

If Bridges looks comfortable in his Texas Ranger gear – the cowboy boots, the cowboy hat, the silver badge – it could be because he got some advice from a real veteran, Joaquin Jackson, author of “One Ranger: A Memoir.” Or because screenwriter Sheridan based the character of Marcus on his uncle, a U.S. marshal from Texas. (“Anytime the writer deeply knows the subject matter, that helps me a lot.”)

But maybe it’s because Bridges, who grew up in Hollywood, the son of “Sea Hunt” series star Lloyd Bridges, has more than a few westerns, old-time and contemporary, under his belt: “Rancho Deluxe,” “Hearts of the West” (both 1975), the infamous epic “Heaven’s Gate” (1980), “Wild Bill” (1995), “True Grit{ (2010) – an Oscar nomination for the latter, as the ornery Rooster Cogburn, a role John Wayne first made famous.

“My father, I used to love it whenever he’d come home from doing a cowboy picture,” Bridges says. “I remember he did a movie called “The Tall Texan,” which is one of my favorites. And he was in “High Noon.” I always liked when my dad did westerns; he did ’em well. He used to love to ride and all that stuff.”

Bridges, a songwriter, a guitar player, a pretty good crooner, spends his time between movie projects touring and recording. His band is called the Abiders, named for the signature line “the Dude abides” from Bridges’ cult classic “The Big Lebowski.”

“We do a lot of songs from ‘Crazy Heart,'” Bridges says. “I’ve got a few albums now, and I do the choicest songs from the albums, songs that I’ve written, but also my dear friend John Goodwin, a Nashville songwriter, we do his songs, as well. And a couple of covers. …We do some Credence, and I’m a fan of a guy named Greg Brown. Do you know his music?

“We do Tom Waits, we do some Byrds. I’m a Byrds fan.”

On the film side, Bridges has a smallish part in the Kingsman sequel, “Golden Circle,” coming in June. And a key role in the real-life firefighting drama “Granite Mountain,” also for next year.

“One of my things when I was starting out, I took a tip from my father,” says Bridges. “He had such success with his TV series in the ’50s, “Sea Hunt.” Skin-diving. He played that so well and developed such a strong persona as a skin diver that people thought that that’s what he was, and he got offered a lot of skin-diving roles.

“And here’s a guy who was a Shakespearean actor, who replaced Richard Kiley on Broadway in “Man of La Mancha.” You know, he could do all kinds of things. But I saw the frustration that he had, tied to this one strong persona. And that’s just a very common thing that happens in movies. You do one part well, and then you get offered a lot more of the same thing.

“I’m trying not to let that happen to me.”