“How the heck did he get on that island? How long has he been there? How the heck did he get cigarettes? How did he get Beanie-Weenies?”

Those are just a few of the urgent questions Matthew McConaughey had for Jeff Nichols once the actor had pored over the writer / director’s screenplay, “Mud,” and signed on to play the title role — a fugitive wanted for murder hiding out on a deserted island in the middle of the Mississippi, down Arkansas way.

In “Mud,” McConaughey gives another in a string of remarkable — and remarkably off-center — performances. Since starring in “The Lincoln Lawyer,” the 2005 thriller adapted from Michael Connelly’s bestseller, the Texas actor has put on a Stetson and a smirk to star as a ruthless D.A. in the dark true-life comedy “Bernie.” He was a seriously twisted lawman in “Killer Joe” — a Texas Ranger who moonlights as a hit man and makes Gina Gershon do unseemly things with a chicken bone. In “The Paperboy,” you don’t even want to know what happens to the Florida newspaper reporter McConaughey plays. Suffice to say it’s a Lee Daniels movie — the guy who brought us that trippy portrait of an obese 16-year-old black girl, “Precious.”

Then came “Magic Mike” — McConaughey in a G-string, grinnin’ and gyratin’ at a strip club featuring buff, baby-oiled male dancers.

So what happened to that other Matthew McConaughey, the one who cruised through instantly forgettable Hollywood rom-coms opposite leading ladies like Sarah Jessica Parker, Kate Hudson and, um, Kate Hudson?

“It was a purposeful sort of redirection,” McConaughey says, on the phone from New Orleans, reflecting on his recent career choices. “I didn’t say, ‘This is what I want to do.’ What I did say was, ‘You know what? I want roles where I’m a little shaky in my shoes.”‘

Forget about those projects McConaughey knew he “could do tomorrow, or next week” — or do in his sleep.

“There’s nothing wrong with that,” he says. “But I wanted something that makes you go, ‘Whoa! I’m not sure what I’m going to do with that, but I sure am turned on by it. And then: Let’s jump off into the abyss and find out.”‘

“Mud” is one of those somethings that compelled McConaughey to jump into the abyss. Nichols — the Arkansas filmmaker whose 2011 film, “Take Shelter,” channeled a post-9/11 collective anxiety into the character of a husband and father (Michael Shannon) who may or may not be going mad — wrote “Mud” with McConaughey in mind. The man McConaughey plays — tattooed, talking mumbo jumbo — has returned to his childhood home after killing a guy who was abusing the girl he loves (an almost unrecognizable, and unrecognizably good, Reese Witherspoon). And then these two kids, middle school buddies (Tye Sheridan and Jacob Lofland), discover Mud living in the trees. Imagine a cross of “Huckleberry Finn,” “Stand By Me” and “The Night of the Hunter.”

“It had those mythical Southern elements,” McConaughey says of Nichols’ script. “You’re on the Mississippi River, there’s soothsaying and superstition …

“And you ask: How far will this man go for love? And then there is this kid, (Sheridan’s) Ellis, who the story is really about,” he explains. “We’ve all had that time around his age where our dreams, our romanticized views of love, are banging against the ceiling of reality. And reality is just nailing us.”

McConaughey’s Mud is grown up, of course, but his views about love — and about the woman played by Witherspoon — aren’t exactly rooted in the real world, either.

“However old Mud is,” the actor says, “he has never come out of the clouds, never come back to reality, and won’t let the world teach him better or teach him worse — however you want to look at it.”

Things have been going nonstop for McConaughey. He shot “The Dallas Buyer’s Club,” losing more than 35 pounds to play a man with HIV. He’s in New Orleans shooting “True Detective,” a limited-run TV series with Woody Harrelson. McConaughey worked with Martin Scorsese on “The Wolf of Wall Street” (yes, Leonardo DiCaprio has the lead), and has signed up for a time-travel film, “Interstellar,” with Christopher Nolan (“Inception”) at the controls.

“I’ve never had any complacency,” the actor says. “In choosing these roles, I didn’t have one day of feeling like, ‘Oh, I know what this is about, I know who this guy is.’

“Every day I was thinking, ‘OK, how do we make this really true today? Not better, not more funny, not more dramatic — just how do we make it more true?’ And that’s a great question to ask, as an actor.

“And it’s a good way to sum up the roles I’ve been choosing. … I’ve been asking that question all the time: What’s the truth of this character?”