Jamie Cullum’s new recording is called The Pursuit, so it’s fair to ask what the pianist and singer is chasing. A reasonable answer: He’s chasing your admiration, your tapped toe, your spreading smile.

His first mature recordings, Pointless Nostalgic (2003) and Twentysomething (2004), put Cullum in the Harry Connick, Jr. category, mostly recording jazz vocal standards with a modern swagger and serious improvising piano chops. But they also the showed Cullum covering Radiohead (“High and Dry”), Hendrix (“Wind Cries Mary”), and Jeff Buckley (“Lover, You Should Have Come Over”). This kid was cheekier and hipper than Connick—a bit of a modern rocker when he wanted to be, but also a singer who could put across stuff like “Old Devil Moon” and “I Get a Kick Out of You” without seeming like a mere Sinatra, Jr.

The question is who this guy is going to become. Diana Krall, for example, made her album of non-jazz originals (The Girl in the Other Room), but ultimately abandoned becoming a singer of new, contemparary songs. She is still a powerfully popular jazz singer, but that’s what she is: a jazz singer.

Cullum’s 2005 disc, Catching Tales, clearly tilted his work in favor of original material that was not necessarily bounded by the conventions of “jazz”. This wasn’t a lurch for crossover appeal, as Cullum had been infusing all his music with this kind of energy from the start. But now it was clear that he didn’t plan to retreat. Tales sounded great, but the different sounds on the varying tunes made it seem a bit stitched together. Cullum wasn’t going to do just one thing, fair enough, but the different things didn’t sound like they came from a single idea.

But five years of touring and playing make the new Jamie Cullum album more convincing. The Pursuit is the same but better. The same but pretty darn wonderful, in fact. He starts with a standard (“Just One of Those Things”) that is driven by swinging piano and big band horn jabs, but as the pop (and mostly original) material takes over, it is notable that the basic sound of Cullum’s band doesn’t change: acoustic piano, horns, a loosely grooving rhythm section. The feeling moves into pop and rock territory, but all the music is clearly coming from one guy, one sensibility. With The Pursuit, Jamie Cullum may be making insistently pleasing, likable music, but it is not calculated or too easy. This is simply a clear and driving sensibility, an omnivorous way of making music. And it works.

The bulk of the material here is highly melodic original music. “I’m All Over It” is a jaunty breakup song that skips deliciously. “She’s a melody / That I try to forget but I can’t / Still follows me / When I wake in the dead of the night / And know that I can’t fight that song going round in my head.”  The lyric aptly describes how this tune works on your noggin, nicely lodging there, with Cullum’s roundhouse piano moving the tune forward with a sloppy fun.

“Wheels” takes a skittering double-time train beat on brushes and a five-note piano lick and sets up a floating anthem with cheekily grim lyrics. Could it be on a pop radio station in 2010?  Almost, but it’s probably a little too good. “Love Ain’t Gonna Let You Down” is a beautiful ballad, but one in a grooving 6/8. This is the kind of contemporary pop music that could almost be coming from John Mayer or Jason Mraz, except that it is more harmonically lush and it uses both Cullum’s piano and the horn section to layer the simple melody in the texture of another era.

Cullum, without making a big deal of it, has a knack for bringing forward some of what was great about the past without seeming like he is from the past. “You and Me Are Gone”, for example, evokes the kind of hip jungle-Latin groove that was the terrain of Louis Prima in the 1950s and uses a pre-chorus with a few Steely Dan-ish jazz chords. But, ultimately, it’s not absurd to imagine an American Idol singer putting this across as a hit. This is exuberant music that shouldn’t seem too jazzy for radio listeners, even if Cullum does include a lightening piano solo in the middle.

There are a couple of intriguing outliers on The Pursuit that could make or break the disc for some listeners. When I played Cullum’s cover of Rihanna’s “Don’t Stop the Music” for my 20-year-old daughter, she just wasn’t buying it. Cullum replaces the original’s electro-house groove with a sinuous acoustic bass and brushed drum kit, with his own piano building the groove from quiet to loud—as if the tune had been conceived by Herbie Hancock in the 1960s. I love it, but I’m an old guy undistracted by the original’s appeal.

That same daughter, however, loved Cullum’s “Not While I’m Around”, a grooved-up version of the plaintive Sondheim song from Sweeney Todd (and she knows the “original” version as well as I do). Again, Cullum’s jazz trio approach is part Hancock and part Radiohead, and his comfort with harmonic complexity and small dashes of improvisation makes the production feel fresh while his sense of drama builds the performance to a soaring climax. I’m not as keen on Leslie Bricusse’s “If I Ruled the World”, where the Broadway melody is played fairly straight except for some synthesized bric-a-brac that I suspect will sound dated in a few years. The disc’s weakest tune, “Music Is Through”, similarly trades in little snippets of modern production (synth bass) that are neither truly up-to-date nor the kind of modern-noir that is Cullum’s heart and soul.

Those exceptions, however, are blips on the surging success of The Pursuit. Jamie Cullum isn’t going to get trapped in the past, even as he straps parts of the older styles onto his back. He is moving forward, eager for you to listen and enjoy but somehow not encumbered by a sense of “selling out” for popularity. It’s a fancy trick, and he deserves to bring a healthy audience along for the ride.

As with so much other smart pop music these days, you worry that the commercial infrastructure that can get this music to its audience isn’t out there. My recommendation: buy this disc for a friend and let word of mouth (or is that now called “Facebook”?) do its thing. Jamie Cullum ought to be a phenomenon. He is already a success.