Headlining Hollywood’s Fonda Theatre Thursday night, Margo Price initially appeared in a dark fringe jacket with just enough Nudie-style, flowery embroidery on the back to hint at a debt to the SoCal country-rock ethos of Gram Parsons, who died a decade before she was born. About two-thirds of the way into her enormously engaging set, she whipped off that jacket to reveal a jersey emblazoned with the rhetorical question “What Would Dolly Do,” reinforcing that she’s a Nashville neo-traditionalist first, and a skilled Joshua Tree wanna-be secondarily, perhaps.

And then the first thing she did after removing that jacket was something Dolly definitely would not do (nor Gram, probably, either): get behind a drum kit to kick some ass for an extended rock and roll instrumental package. That’s something .38 Special might do, but it worked for her, adding an extra dose of dynamism to a show that’d already had Price playing acoustic and electric rhythm guitars and piano, as well as just putting everything down to prowl the stage while her expert five-piece band took care of honky-tonk business. Price is one performer who’ll never need to be nagged to “step up.”

She’s making the national rounds on the coattails of “All American Made,” her second excellent long-player on the Third Man label, where she’s the most successful signee not to have Jack White as an actual member. The fullness of her 105-minute set made it feel like she’s been around longer, but it wasn’t hard for Price to fill out the length of a more longstanding veteran’s show with the bulk of the two albums, plus a few well-chosen covers: an especially greasy version of “Most Likely You Go Your Way and I’ll Go Mine,” with the steel guitar working in slick tandem with the lead player on Bob Dylan’s classic riff… an encore version of “Proud Mary,” Ike and Tina-style… and a medley of Merle Haggard’s “I Think I’ll Just Stay Here and Drink” and Willie Nelson’s “Whiskey River,” interpolated into her own signature drinking song, “Hurtin’ (On the Bottle).”

With her sophomore album last fall, Price expanded her songwriting palette to include more overtly socially conscious material. While it might seem like a stretch to easily position those against a song like “Hurtin’,” she does have some models for that… in Haggard and Nelson themselves, for starters. It wasn’t a let’s-slow-things-down kind of concert, but Price did sit alone at the piano for the title song of “All American Made,” a sprawling national-mood commentary that somehow manages to invoke both Tom Petty and Oliver North. She also kept things mellow in premiering one brand new song, “The Devil’s in the Details,” a broad-ranging mixture of the personal (“I got crooked teeth but true intentions/A real sound mind with slight dementia”), the political (“They own the pipelines and avoid paying taxes/The water gets poisoned while the president relaxes”), and the religious, or non-religious (“If you turn your back on Jesus, do you really think he cares?”).

But for all of those expansive songwriting aspirations, Price still aims to front a glorious country roadhouse band, before she does anything else. (The ideal place to see them might be actual roadhouse Pappy and Harriet’s, out in Pioneertown, where they’re playing a bonus show Saturday night. Don’t bother trying to get tickets.) Price is never so egotistical about her winsome wail that she doesn’t seem to be affording nearly equal time to lead guitar, steel, slide, and piano solos — all of which finally get more than equal time in the extended, hyperactive set closer, “Paper Cowboy”… a B-side that is definitely the elevated A-side of any of her live shows. She stays in good-time mode while veering into or outside the margins of country music: “Do Right By Me” has the gospel/roots feel of vintage Delaney & Bonnie, while the rock instrumental theatrics that provide an extended coda to “Cocaine Cowboys” (one of those tunes that had Price on drums) could be out of the catalog of Delaney & Bonnie’s old pals, Derek & the Dominos.

But when it comes down to it, her Dolly jersey nod notwithstanding, “What Would Willie Do” is as clear an underlying maxim as she has, even if she thrashes her hair around a little harder than Nelson has ever been known to rock his ponytail. The new song, “Devil’s in the Details, had her husband, Jeremy Ivey, playing a harmonica solo right out of the Mickey Raphael handbook. And the connection was rather obviously cemented when she had Willie’s son, Micah Nelson, come out to duet with her on “Learning to Lose,” which his dad sang with her on her latest album. Of course, she was taking fewer cues from Nelson when she was working the stage on “Proud Mary,” but a combination of Willie and Tina is quite a noble thing to go for.