He reveals, however, that the moustache's days are (at least temporarily) numbered.
"When all this is all over, I'm going to shave it off," he says. "I can go places and not be recognised when I don't have a moustache and I manage to keep my mouth shut."
By "all this", Elliott means his current glut of work, 51 years into a career that hasn't slowed yet.
His Oscar-nominated supporting turn in A Star Is Born as Bradley Cooper's brother/tour-manager Bobby Maine has been the high point of a current hot streak that has also taken in two Netflix series, The Ranch and Grace and Frankie; an elegiac semi-autobiographical feature called The Hero; numerous voice roles; and, now, an offbeat independent number whose title you can't fully savour until you've heard Elliott growl it aloud.
"The Man Who Killed Hitler and Then the Bigfoot," he rumbles, recalling the day his agent handed him the script. "I was very sceptical. Obviously."
But with the screenplay for this odd-and-proud project, by first-time director Robert D Krzykowski, came a note from the great independent film-maker John Sayles, urging Elliott to give it a chance.
And it was this — along with the eyebrow-raising involvement of special effects pioneer Douglas Trumbull, of 2001: A Space Odyssey fame — that persuaded him to take a look.
He liked what he read.
The Man Who Killed Hitler and Then the Bigfoot, is no blood-streaked pastiche, but an earnest rumination on the elderly ex-warrior's lot: a kind of grindhouse Beowulf, in which the hero, Elliott's grizzled US Army operative Calvin Barr, slays two epochal monsters half a lifetime apart.
Aidan Turner, of Poldark fame, plays Calvin's younger self.
Like The Hero, a recent Netflix release in which Elliott plays a western icon meandering through semi-retirement, it feels like the kind of film an actor takes as a way of sizing up his whole life's work.
But Elliott isn't deliberately seeking them out.
"I'm going to be 75 in August, so this is just the story to tell about me," he chuckles.
"It's not that I'm grappling with my destiny as such. But there's a lot to deal with there.
It's a universal theme. It's where we're all ultimately headed."
Besides, he had enough of that on the Oscar campaign trail with A Star Is Born, where his nomination was widely seen as an overdue career-spanning salute.
"I've just spent the last 20 months of my life looking back on all of it," he says.
For him, Cooper's fourth version of the classic Hollywood melodrama is "a killer. The best telling of the tale. I've had some really nice opportunities along the way, but none better than that one. When you can trust your director, your fellow actors and the material, odds are you can get in deep enough to tell some sort of truth."
Indeed, the chemistry between Cooper and his co-star Lady Gaga rang so true, rumours flared that the pair's on-screen romance had spilt into real life — not least after their bottled-lightning duet at the Oscars, performing the film's signature song Shallow at a piano with much reciprocal nuzzling.
Elliott's gruff verdict?
"It's bulls***. I was there through the long slog of awards season, doing Q&As, having meals with them, talking to them, seeing them interact. And what I truly believe is that what you saw of that relationship was on the screen.
"I don't think there was something else going on."
This is an area in which he has some expertise, having met his wife, the actress Katharine Ross, on the set of the 1978 horror film The Legacy. He was single, while she was married to Italian-American actor Gaetano Lisi; Ross's marriage ended the following year.
Elliott describes screen romances as "a fine line".
"It's what you allow, how far do you allow that chemistry to take you. "The reality and the acting, they become one on some level when you get in deep. Other actors say they turn it on and turn it off, and I think, 'Good for you'. It's not easy to do that.
"It's probably one of the reasons there are so many short-lived marriages in the business."
He and Ross became an item soon after her divorce, wed in 1984, and have been together since.
"People ask us all the time what our secret is," he says.
"It's that we love each other and we want to be together. Pretty simple."
- Telegraph Media Group
