The comedian's stint as Jon Stewart's Daily Show stand-in has been consistently funny – but has it been good enough to give him a shot at the job whenever Stewart steps down?

John Oliver's three-month stint hosting America's biggest cable talkshow began positively back in June. The New York Observer hailed it "the best thing to happen to late night since Colbert". Salon asked: "Jon Stewart who?" The Guardian labelled it "triumphant". It would seem that the British comedian has effortlessly slipped into the vacancy created by Stewart's directing sabbatical.

A highlight from recent weeks has been the critique of CNN's new morning show New Day, and particularly its wildly misjudged feature Rock Block, which takes inane to new levels, forcing reporters to deliver serious stories over invasive pop music. In Oliver's words: "You've got the worst of both worlds. You've sucked the fun out of rock, and the weight out of news."

Coverage of the Zimmerman trial was again testament to Oliver's success. The branding of Florida as "the Worst State" following the not-guilty verdict was the ultimate bitter punchline delivered with suitable bravado. Oliver clearly has a passion for reporting the news with a satirical edge – it has been evident in his many pre-Daily Show stints, on Mock the Week and the Radio 4 show he hosted with Andy Zaltzman, Political Animal.

The public have largely welcomed the change from Jon to John. Ratings for the show are roughly comparable to those of last summer. However, there are a few areas to be addressed before Oliver can be considered a frontrunner for securing the keys to the nightly satire, once Stewart has hung his up – not that, at present, there's any indication he's planning to do so.

The need to ensure Stewart has nothing to worry about once he returns in September has left Oliver with a bit of a poisoned chalice. The status quo has remained where elements could have been refreshed ,and Oliver's stand-up ability capitalised on. Mentioning Stewart at the start of every episode only rubber-stamps the fact that Oliver is limited in what he can do. Even the graphics have failed to reflect the interim host.

Then there's the interview slot. Oliver doesn't yet seem comfortable in a one-on-one setting. A recent talk with screenwriter Aaron Sorkin – admittedly conducted in the dark following a power outage – fell flat. But the quality of some guests, and Oliver's pure untethered enthusiasm, is beginning to shine through. Louis CK gave a welcome appearance this week, announcing – to the joy of headline writers – that Rolling Stone magazine, "put that kid on the cover that blew up Boston, so fuck them". Last month's interview with Oliver's comedy idol, Armando Iannucci, resulted in an endearingly overwhelmed host.

When it comes to the Brit covering his birthplace, there's a sense of jubilance. Cue CNN contributor Victoria Arbiter commenting that the Duchess of Cambridge was a "brilliant" royal for producing a boy on her first shot. Oliver, pondering Arbiter's reaction if a girl had been born, replied: "Damnation upon your cursed womb, Catherine! Burn the princess for she has produced a baby of the weaker sex!"

The Daily Show with John Oliver continues to be witty, intelligent and consistently funny. Classic moments have been added to the show's canon, and the ship has been steadied for what could have been a turbulent summer.

• British viewers can watch The Daily Show weeknights at 11pm on Comedy Central Extra.