Wedding military hardware and a martial mind-set to paranoia about global pandemics, “The Last Ship” is an odd duck — a post-apocalyptic tale with a glimmer of hope, tonally similar to (if not as good as) the late, lamented “Jericho.” It’s left to a hardy naval crew and one inordinately beautiful scientist try to save humanity, without knowing exactly what’s left of it to save. Eric Dane is firmly in square-jawed John Wayne territory as the ship’s captain, on a TNT summer drama with plenty of possibilities but also numerous pitfalls as it seeks to navigate rather treacherous narrative currents.

The USS Nathan James has been off in the Arctic and out of radio contact while a devastating “virus of unknown origin” has decimated humanity. But it turns out the ship was on a secret mission, hosting a paleomicrobiologist (no, really, look it up), Dr. Rachel Scott (Rhona Mitra), who is seeking a potential cure.

With governments falling, the disease extremely contagious and little way of knowing who has survived back home, the ship’s steadfast crew (a fairly nondescript lot, other than Dane and Adam Baldwin as his second in command) must battle on, trying to buy Scott time to conjure a vaccine.

Although Jonathan Mostow (“U-571”) directed the fast-paced pilot written by Hank Steinberg and Steven Kane (and adapted from William Brinkley’s novel), the show actually doesn’t hit its stride until the third episode. In that hour, the ship encounters a Russian vessel with nefarious aims (we’re told the Russian government has disbanded), and pivots from something akin to “On the Beach” to “The Hunt for Red October.”

While the Navy’s cooperation on the show — allowing the producers to shoot aboard working destroyers — gives the series an impressive real-world backdrop beyond the CGI, “The Last Ship” will ultimately prove its seaworthiness by developing characters worth spending time with in these semi-claustrophobic surroundings.

Yet beyond Dane, Mitra and Baldwin, the initial side-stories fall pretty flat, in part because the cast is saddled with a lot of tech-heavy jargon, while frequently shouting dialogue through disease-control suits. And frankly, even with the soapy elements that emerge, the show’s creatives seem to have missed an opportunity for occasionally turning things into the SS McSteamy, with Dane’s Captain Chandler saddled with a backstory that has him fretting about his wife and kids.

TNT is certainly launching “The Last Ship” with helpful promotional fanfare, from an ad blitz during the NBA playoffs to pairing it with the summer hit “Falling Skies,” which also happens to involve a world turned upside down.

So as starts go, this one picks up speed, but still feels a little rocky. That said, there’s enough here to want to hang around for a spell, waiting to see whether this crew can find its sea legs — and what dangers lurk just over the horizon.