For the makers of a TV drama, there are two potential disasters at the end of a first season: not being renewed, and being renewed. Disappointing as cancellation is, it at least leaves the possibility that one day the show will come to be regarded as an unfairly truncated lost classic. But get a second run and screw it up, and the successful first year will be weakened by the failure of the next.

But, in approaching Difficult Second Series Syndrome, Billions has more going for it than most contenders. The Showtime series – which returned in the US on Sunday and resumes in the UK on Tuesday – started from a scenario that has long and energetic legs.

Bearded, damaged and bordering insanity … Giamatti as Chuck Rhoades. Photograph: Jeff Neumann/Showtime

The mutual obsession with each other’s destruction between US Attorney Chuck Rhoades (Paul Giamatti) and hedge fund maverick Bobby Axelrod (Damian Lewis) left many places to go and grow from the former’s attempt to charge the financier with corruption in the first run. Manic pursuit has the potential to play out over a long time, which is why Tom & Jerry has survived for almost 80 years and Moby-Dick is 600 pages long.

Cultural allusions are a big part of the Billions shtick, so it seems no accident that Rhoades – bearded, damaged, bordering insanity in his desire to bring his quarry in – has many similarities with Captain Ahab, Herman Melville’s whale-hunter, as well as Ahab’s premier literary descendant, JM Barrie’s crocodile-obsessed Captain Hook. Giamatti would be natural casting for either sailor.

References to the Hanna-Barbera cartoon may be less intentional, although, like the mouse Jerry, Damian Lewis’s mega-rich banker is often, counter-intuitively, the more attractive of the central duo. As in the animation, the viewer is sometimes surprised to feel pleased when the latest hot iron dropped by the enforcer of order lands scaldingly on his own foot while the rodent slithers away.

But, whereas Tom & Jerry came from a TV era in which it was believed that the key to success involved repeating the same situation to infinity, Billions knows it must adapt to survive. Season two cleverly flips the plot, so Axelrod becomes more the hunter, with Rhoades now the one under investigation for inappropriate behaviour.

‘I’m the bull!’ … Damian Lewis as hedge fund maverick Bobby Axelrod with Malin Akerman as Lara Axelrod. Photograph: Jeff Neumann/Showtime

A basic rule of TV survival is the introduction of compelling fresh characters alongside the regulars, and one of the newbies in Billions is Oliver Dake (Christopher Denham), an auditor from the Office of Professional Responsibility. With darting eyes, he arrives in the US Attorney’s New York HQ, seizing documents and testimony suggestive of malpractice. These will be taken to the Attorney General, who has summoned Rhoades to Washington, and seems likely to fire him.

Finding the harpoon aimed at him, Rhoades responds as any self-respecting public official would. He instructs his staff to identify an enemy of the Attorney General, who, if placed under investigation, could give Washington a conflict of interest in sacking him.

Rhoades, with dismay, and the audience, with relish, realise that Dake’s best bet for getting dirt on his target is to persuade Axelrod to go on the record about the stuff that happened in season one. So the investigator soon visits the modernist offices of Axe Capital, where a Henry Moore or Giacometti sculpture can frequently be glimpsed casually in the background, and where Damian Lewis also has a fascinating new character to swear at.

‘The hedge fund industry is under attack!’ Lewis as Axelrod with Taylor (Asia Kate Dillon). Photograph: Jeff Neumann/Showtime

Taylor (Asia Kate Dillon) is a shaven-headed intern whose opening gambit is: “My pronouns are their, they and them.” They rapidly impress Axelrod, who believes their expertise will give an “edge” at a time when “the entire hedge fund industry is under attack!” It’s a mark of Lewis’s charm that he manages to make his mission to save these tax-avoiding economic vandals sound like a fight for right.

Billions continues to have two signature qualities: one verbal, one visual. Key to the way the show looks is a policy of seconding episodes to guest-star directors, often from cinema. Neil La Bute, Susannah White and Stephen Gyllenhaal were hired in the first season, and the third instalment of season two marks a sabbatical for the documentary-maker Alex Gibney. The risk of this approach is the lack of a controlling vision, but the benefit is that the eye is kept alert for changes of style, like walking through an art gallery.

Key to the way the show sounds is the insistence on talk so fast and so smart that it makes the characters in The West Wing sound monosyllabic. A particular trick of the script is imagery from the natural world. Lead writers Brian Koppelman and David Levien (who created the show with financial journalist Andrew Ross Sorkin) seem to have spent their formative years watching David Attenborough documentaries.

‘I’m like a goddamn octopus!’ … Oliver Dake (Christopher Denham) arrives to wreak havoc for Chuck Rhoades. Photograph: Jeff Neumann/Showtime

“I’m like a goddam octopus!” warns Dake, while Axelrod declares: “I’m the bull!” Rhoades, though, is a different kind of animal: “Don’t serve rabbit food to an elephant and ask if he’s full!” There are even beastly metaphors for whole nations – “China’s a pig on LSD … you never know which way it’s gonna run!”

The best lines still tend to go to the guys, although Maggie Siff, as psychiatrist Wendy Rhoades, gets a welcome couple of scene-winning smashes this time, especially in a set-piece speech to a conference of hedge-fund managers.

The torrent of pop culture references continues. Before watching season two, it is wise to be up to speed on, among other things, The Philadelphia Story, the Dalai Lama, the musical Hamilton, Ali G, the Roman poet Pindar, and always, not far below the surface, Moby-Dick.

More than a century and a half after Melville’s novel was published, it is not much of a spoiler to say that, when Ahab finally thinks the quarry is in his grasp, he is outmanoeuvred and drowned. Constantly floating this possibility in front of viewers, Billions shows signs, like the book, of becoming more interesting the longer it goes on.

Billions is on Sky Atlantic from 21 February at 9pm, and continues in the US on Showtime on Sundays at 10pm.