People aged 32 and 33 in England are now being invited to book their first dose of the coronavirus vaccine.
The NHS opened bookings to about one million people on Saturday morning.
It comes as surge testing is being rolled out in North Tyneside and parts of west London affected by the variant first found in India.
Meanwhile, Germany says anyone arriving from the UK will have to quarantine for two weeks, amid concerns over the spread of the variant.
The variant - also known as B.1.617.2 - is responsible for the majority of new cases in pockets of England.
People in North Tyneside are being urged to get tested amid a rise in cases linked to the variant. Two extra mobile testing units in North Shields and Wallsend opened on Saturday.
Wider surge testing is also being rolled out in Ealing, Hillingdon, Brent, Harrow and Hounslow. It will target secondary schools in the five boroughs and PCR tests will be offered to everyone else in the area who do not have any symptoms.
More than 37.5 million people in the UK have now received their first dose of the jab - and 21.6 million second doses have been given.
Text message invitations are being sent out to 33-year-olds on Saturday and to 32-year-olds on Monday.
They will appear as a message from "NHSvaccine", and people who cannot go online can call 119 to book a jab
The other UK nations are already offering jabs to younger age groups - people aged 30 and over are eligible in Scotland, as are over-18s in Wales and over-25s in Northern Ireland.
More than 40% of adults in England have now had both doses of the vaccine - meaning they have maximum protection from the virus.
And among those aged 35 to 39, more than half have had their first jab since becoming eligible earlier this month.
Health Secretary Matt Hancock said the government was still on track to offer a vaccine to everyone by July.
People in their 30s and pregnant women are being offered either the Pfizer-BioNTech or the Moderna vaccines if they are available, rather than the Oxford-AstraZeneca jab, because of concerns about a possible link between the Oxford jab and rare blood clots.
And over-50s and those considered clinically vulnerable will have their second doses brought forward to eight weeks after their first, rather than 12, following concerns about the Indian variant.
The guidelines from the Joint Committee on Vaccination and Immunisation (JCVI) say there is some flexibility in the way the rollout is conducted at a local level.
Bolton continued to be the worst-affected area in the country with 385 cases per 100,000.
And people aged over 18 in parts of neighbouring Manchester have been urged to book vaccinations.
Rochdale also said it would offer jabs to some over-18s.
Andrea Fallon, Rochdale's director of public health, told BBC Radio 4's Today programme the council was focusing on the priority groups as well opening five drop-in centres as part of a "new programme" this weekend.
Government figures on Friday showed there had been another 2,829 confirmed cases of the virus in the UK, and a further nine deaths.
There were also early signs of a "potential increase" in Covid infections in England, the Office for National Statistics said on Friday - following a fall for five straight weeks.
Although the UK government is still advising against non-essential travel to most countries, some European nations are indicating whether UK tourists will be allowed in.
Spain is officially lifting restrictions for UK tourists from Monday, and arrivals will not need a PCR test.
Heathrow Airport has decided to open a separate terminal just for passengers arriving from "red list" countries, after concerns that travellers from red countries were queuing up with people from destinations with lower instances of Covid.
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