Mr Hancock hailed the start of the rollout of the Oxford/AstraZeneca vaccine as an 82-year-old former maintenance manager became the first person to receive the jab outside of clinical trials.
He said the NHS has the capacity to deliver two million doses a week of the vaccine once it receives supplies from the manufacturers.
But with the latest data showing a 33% rise in the number of confirmed coronavirus patients in hospital in England between Christmas Day and January 2, he warned there would be “some very difficult weeks” to come.
Asked about the prospect of another national lockdown, he acknowledged that the current restrictions are insufficient to control the spread of the disease.
He told Sky News: “We don’t rule anything out, and we’ve shown repeatedly that we will look at the public health advice and we will take the public health advice in terms of what is needed to control the spread of the disease.
“This new variant is much easier to catch, it is much more transmissible, and we’re now seeing the effect of that in lots of different parts of the country, unfortunately.
“And it means that, whereas the old Tier 3 was able to contain the old variant, that is proving increasingly difficult in all parts of the country.”
His warning came as the Scottish Parliament was being recalled to consider further restrictions following a rapid increase in cases there.
Despite the concerns, Mr Hancock insisted it is safe for primary schools to reopen in all but the worst-hit areas of England following the Christmas break.
He said teachers are at no greater risk of contracting the disease than the rest of the population.
“There is clear public health advice behind the position that we have taken and that is what people should follow because, of course, education is very important as well, especially for people’s long-term health,” he said.
However, Prime Minister Boris Johnson is coming under pressure from unions in the education sector to order a “pause” in a return to the classroom until the safety of staff and pupils can be guaranteed.
In a joint statement, the GMB, NAHT, NASUWT, NEU, Unison and Unite unions said there is a “serious risk” of staff falling ill while the rate of infection is so high.
“The Government’s chaotic handling of the opening of schools has caused confusion for teachers, school staff and parents alike,” they said.
“Bringing all pupils back into classrooms while the rate of infection is so high is exposing education sector workers to serious risk of ill-health and could fuel the pandemic.”
For Labour, shadow education secretary Kate Green said there needs to be a “stronger set” of coronavirus restrictions in place with a clear “stay at home” message to the public.
She told BBC Radio 4’s Today programme: “It is very clear that the Government has lost control of the virus, we’re seeing a really alarming rise in cases and in the spread of the infection.”
Meanwhile, Professor Stephen Powis, national medical director of NHS England, described the rollout of the Oxford/AstraZeneca vaccine as “another turning point in our way out of this pandemic”.
Dialysis patient Brian Pinker received his jab at 7.30am on Monday from nurse Sam Foster at Oxford University Hospitals NHS Foundation Trust’s Churchill Hospital.
Prof Powis said it followed months of preparation by the NHS for what will be “the biggest vaccination programme in our history”.
“We’ve already delivered over a million vaccines of the Pfizer jab; now we’ve got the AstraZeneca one, so we aim to get it into people’s arms as quickly as it is supplied to us,” he told BBC Breakfast.
“If we get two million doses a week, our aim is to get two million doses into the arms of those priority groups.
“As the Prime Minister said, we’re aiming for tens of millions of doses by the time we get to April.”
He said the vaccine will be delivered in around 100 hospital hubs and 700 centres in GP practices and in the community by the end of the week, with plans to expand as more supplies become available.
At the same time, Mr Hancock said the “bureaucracy” involved in signing up to be a volunteer vaccinator is being reduced to make it easier to people such as retired doctors to enlist.
“We’re going through the different parts of that process to streamline it as much as possible but again that isn’t the rate-limiting step,” he told BBC Breakfast.
“Because, at the moment, the NHS, with the people that it has got already, is able to deliver the vaccine as it can be produced, but obviously I want to make that easier.”