Exhaling with relief, I stepped on to the dock. I’d just returned from an excursion 11 miles offshore in an inflatable boat, hot on the trail of a whale. The whale never surfaced, but my fear of the water certainly did as we paused near a large patch illuminated by the sun on this otherwise cloudy day.

“See the Cory’s shearwaters up there?” marine biologist Adriana Gouveia asked, pointing to several large birds gliding on the wind.

“That’s how we can tell this is a feeding ground.”

These were the birds I’d heard from my hotel room the night before; their cackling calls so distinct I grabbed my phone and scrolled in the dark until I could identify the source. The next morning, I envied their wings. Being tossed about by sea swells stirred both motion sickness and my general aversion to deep water.

White-knuckling the seat in front of me, I politely requested a return to shore, but my morning with OceanEmotion hadn’t been fruitless – a pod of common dolphins came to investigate our small craft. They swam close enough for me to hear the puffs of air from their blowholes.

The Azores, a Portuguese archipelago about 930 miles off the coast of the mainland, are revered for their biodiversity. Nearly 11,500 land and marine species are on record here. With nine verdant ­islands offering swimmer-friendly beaches, hiking trails and natural hot springs, it’s a holidaymaker’s paradise.

There is so much on offer, in fact, that Azores Airlines reports carrying 1.4 million passengers last year – a 52 per cent increase from pre-pandemic 2019. Records are expected to break once again ­during this year’s peak travel ­season, with August being the busiest month.

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These islands welcomed more than 24,000 UK travellers last year. Samantha Bosman, senior product executive for Portugal at tour operator Audley Travel, says that “both enquiries and bookings for the Azores have grown since we launched the destination in 2019”. This is partly because the number of direct flights to the islands has increased.

The archipelago is often referred to as the Hawaii of Europe. I don’t like declaring one place to be the regional version of another (spare me your “Venice of the North”) – and most Britons will never have seen Hawaii, which is a 17-hour-plus flight from the UK. But touching down on São Miguel, I understood how you could draw those parallels. With its densely forested hillsides that rise straight from teal waters, the island is visually spectacular.

The undulant, volcanic landscape felt especially pronounced as I arrived in the crater of Furnas. There are natural hot springs and boiling calderas from which steam billows round-the-clock – it’s easy to believe that 50 per cent of São Miguel’s power comes from geothermal energy.

The sulphuric plumes nestled into the fibres of my clothing, not unlike when I soaked myself in the thermal pools at Octant Furnas shortly after check-in. I happily dog-paddled round the murky, mineral-rich soup before emerging for a stroll through the village.

Furnas is tiny, and everyone is either dressed as though they’re returning from a hike or on their way to the spa. The laid-back feel suited me fine as I made my way to Queijaria Furnense, a cheese shop and restaurant showcasing São Miguel’s finest dairy products. Approximately 30 per cent of Portugal’s milk production happens in the Azores. All that fresh air and juicy grass does make for delicious cheese (and beef).

I had to leave São Miguel before I could sink my boots into the island’s dozens of hiking trails. But I soon clocked a breezy 10,000 steps around of Angra do Heroísmo, capital of the neighbouring island of Terceira (Azores Airlines offers flights between the two, which take 40 minutes).

The Unesco-listed seaside city feels plucked from Portugal’s mainland, with its intricate tile designs and lively restaurants where a cracking piece of tuna and a ­negroni can be had for less than £30. The local pastries at Queijadas da Dona Amélia are another highlight. The warmly spiced, ­ultra-chewy tarts could usurp Portugal’s custard variety, pastéis de nata.

Just 55,000 people live on Terceira and the 150-square-mile island is criss-crossed with low-slung stone walls that serve as livestock fencing and property borders. Residents – who mostly live near the coastline to preserve the natural beauty of the interior – are fond of reminding guests that each of the Azores islands are unique and worth a visit.

Terceira is the whale-watching hot spot, though there are excellent walks with spacious, quiet trails (the views atop Monte Brasil are worth the trek) and horse riding for those who prefer to stay on dry land. But with just two cities and a smattering of villages, could the general feeling towards increasing visitor numbers sour?

“I’m not worried about the tourism in Terceira,” said Ines Meneses, a resident. “Terceira is very into our hospitality and events. People on other islands may feel differently, but my only concern here is about rising real estate prices.”

Another local, Paulo Matos, said he welcomes the practical advantages.

“Stores aren’t always open for tourists; people come on Sundays and find everything is closed. Businesses aren’t really thinking about that – more visitors could be beneficial.”

Now, I’m convinced, is the time to go to the Azores – while it’s still quieter than Madeira, free from the protests against mass tourism on Spain’s islands and, crucially, still budget-friendly.

Rooms can be booked for under £100 a night in a four-star hotel, beer is less than £4 a pint and the locals are happy to have us.

Summer Rylander was hosted by Audley Travel, which offers an eight-day tailor-made trip to the Azores from £3,950pp (based on two travelling), including flights, accommodation on a B&B basis, transfers and excursions; audleytravel.com.

Getting there
Azores Airlines and Ryanair offer direct flights between London and Ponta Delgada.

Inter-island flights are best serviced by Azores Airlines.

Villages are walkable, but public transit on the islands is limited. Hiring a car is recommended and vehicles are widely available.

Staying there
Octant Furnas has doubles, with breakfast, from £160; furnas.octanthotels.com
Terceira Mar Hotel has doubles, with breakfast, from £60; bensaudehotels.com

More information
visitazores.com