It is 7.00pm on the southeast corner of the island of Mykonos, the sun is edging drowsily into the sea, and the owner of the brand-new The Wild Hotel by Interni is gesticulating with a cigarette in the direction of a yacht in Kalafatis Bay. Mr Alexandros Varveris is an ebullient Athenian with a long familial connection to the island and he is explaining to me how things used to be here: “You would have Jackie Kennedy and her sister and the heir to the Krupp armament fortune. He would never carry cash and his secretary would just travel round the island at the end of the season asking if he had visited this taverna or that shop. I expect he paid,” he takes a drag of his cigarette, “a lot of extra money.” Mr Varveris is attempting to bring back a little of that bold-face name glamour with two new projects on the island (he has long owned a restaurant, Interni, in the old town, too).

For the past few decades, Mykonos has become synonymous with a certain style of partying: usually boozy, quite often excessive and perhaps not altogether in the Ms Jackie Kennedy mould. And that is no bad thing, by any means. But if you wanted a bit of five-star comfort, a quiet place to lay your head, and you weren’t in possession of a motor yacht or a multi-billion-pound armament’s fortune, then you were a little short of choice. As Mr Varveris says, he wanted to “bring a new dynamic” to the island.

The infinity pool of the 40-room Wild Hotel seems to imply he has done just that. To sit beneath a sunshade, drinking Greek rosé and watching the sun on the Aegean, is to be seduced. It is peaceful; it’s quiet; everything is in a pale shade of terracotta. If you are willing to move, though, there is an outdoor gym, a restaurant, The Taverna, serving the best taramosalata I’ve ever tasted and a pretty private beach accessed by deep zig-zagging steps. What there is not is any hullabaloo, which is the whole point.

If you do wish to stray beyond the confines of the hotel, 15 minutes’ drive from here, at Ftelia Bay, is the second element in Mr Varveris’ plans for the island: Ftelia Beach Club. Beach clubs on the island traditionally follow a similar format: you rent a bed for the day – sometimes at prices of up to €150 – and you spend your day at them, drinking, dancing and perhaps watching a drag performance, depending on your mores.

Arriving at the beach club, the first thing that strikes you is the design. Italian architect Mr Fabrizio Casiraghi – a distant cousin of those Casiraghis – has created a 1960s-style confection that is half Italianate and half French. It’s all stripped woods, earthy reds with endless twisting trellises of flowers threaded hither and thither. It looks like a set by Mr Federico Fellini for the part of a film where everyone falls in love.

My bright, white English skin makes an amusing counterpoint to the bronzed torsos on the beach, but I don’t care because the waiters are quick, the cocktail list long and the water sapphire blue. It is also much more accessibly priced than other beach clubs on the island. After an enormous Greek-style sharing lunch – featuring spectacularly well-cooked local fish and a salad featuring anchovies nearly the size of my palm, I fall into the generous embrace of a large, red, foam sunlounger. And I do something I never do at the beach: I sleep.

Sleep, good food, sharp design and, it seems, rosé wine, are the central strands of Mr Varveris’ Mykonos. When I return home, I have the greatest pleasure of all: denying my friend’s catcall questions as to whether I am feeling “all Mykonos’d out”. On the contrary, I felt full of the joys of spring. If still a little pale.