Badrutt's Palace, St Moritz, Switzerland: Soaking in the summer at Switzerland's glitzy alpine resort

St Moritz's grandest hotel knows how to put on a welcome. Badrutt's Palace, which can be spotted from miles away, thanks to its landmark tower and turrets, has sent a chauffeur to meet me off the train. I imagine that he's outside, waiting beside the Rolls. But no – as my train slides into the station, I spy Giuseppe Pini on the platform holding a timber-framed, tower-topped sign featuring my name, printed out as large as you please in a graceful serif font.

The hotel is renowned for its fleet of Rolls Royce cars, which includes a 1968 Phantom that once belonged to the British royal family. The cars spend all day shuttling guests to and from the station (and Engadin​ airport, 10 minutes' drive away) but they still cause heads to swivel.

Pini swings open the Rolls' rear-hinged back door – commonly known as a suicide door but the manufacturer prefers "coach door", if you please. It means you can pretty much dive straight onto the leather seat. I dive, buckle and settle, waiting for my luggage to be loaded. People on the footpath stare right at me, probably wondering how I managed to score such a luxe ride. Where is one supposed to look while under such scrutiny? C'mon driver, hurry up and get me out of here.

At Badrutt's I'm ushered to a sixth-floor room with lofty views over oval-shaped Lake St Moritz – when frozen, it hosts polo matches and horse races. A humidifier is activated to counteract the dry air at this (1865-metre) altitude and the television is piping a hotel-curated soundtrack, with Louis Armstrong crooning "Heaven, I'm in heaven …" and I can't disagree with that. If I lean out over the balcony and look left, I can just make out the Hotel Muottas Muragl, accessible only by funicular. That's where I intend to head for sunset.

Firstly, I swing through the corridors with the hotel's Lars Wriedt. Between antique chests and paintings is the odd single chair, placed outside some doors for guests' "people" (security guards, nannies, butlers, assistants and the like). We peek into the corner suite that Alfred Hitchcock favoured and another suite that's hiding a safe as big as a suitcase. "It's good if you want to pay in gold bars," jokes Wriedt.

Summer is St Moritz's secondary season; winter is when it really buzzes. That's when you won't find a spare table in Le Grand Hall, the hotel lobby known as "the living room of St Moritz". Staff almost double from 280 in summer to 520 in winter, and the in-house Nobu restaurant, Matsuhisa, opens its doors only for winter.

Still, there are reasons aplenty to hit St Moritz in summer. If I had more time, I'd circumnavigate the lake and soak up the dramatic Engadin Valley scenery, or head to the neighbouring village of Celerina​ to pad over its three barefoot hiking trails. Instead, I'm ducking over to Samedan eight kilometres away (or two train stops) to soak in the country's first vertical spa.