Jeremy Hunt has called for the government to cut the time required between Covid vaccine doses to allow more booster jabs to be given as concern grows about the pace and organisations of the latest rollout.

Speaking in the Commons, the former health secretary said: “If you look at the higher hospitalisations, cases and death rates [in the UK] compared to countries like France and Germany, the heart of it is not actually things like mask-wearing and Covid passports, it is their higher vaccine immunity.”

Hunt said the government should examine cutting the wait after a second jab before people are offered a third. “This decision that’s been made that you can’t have your booster jab until six months after you had your second jab – how hard and fast should that rule be? Does it really matter, when it’s only nine weeks till the Christmas holidays, if someone has a booster jab after five months?”

Booster jabs should be given to over-50s and other eligible groups six months after their second vaccine dose, according to advice from the Joint Committee for Vaccinations and Immunisations.

Answering Hunt, Maggie Throup, the vaccines minister, said that immunity “doesn’t fall off a cliff edge, it has waned slightly … there is still time for people to come forward … They still have a huge amount of immunity over those who have yet to get their first jab.”

The Conservative MP Mark Harper said it was unfair for the government to be warning of new restrictions if booster jab take-up did not improve, saying there were administrative obstacles.

“I can’t see any evidence that the public is not coming forward when asked for booster doses … If actually the slowness of the rollout is to do with either the way the NHS is administering it or the way ministers are administering it, if there is going to be a problem later in the autumn that will be on ministers, not on the public,” he said.

Some 2.68 million people aged 80 and over in England have received two doses of vaccine, of whom 1.34 million (50%) are now estimated to have had their booster dose.

Labour has called on the government to speed up the programme. In the Commons, the shadow health secretary, Jonathan Ashworth, said the NHS was coming under pressure because “the vaccination programme is now stalling” and called on the government to commit to 500,000 jabs a day.

“Ministers cannot blame the public when 2 million haven’t even been invited for a booster jab and on current trends we won’t complete the booster programme until March 2022.”