Speaking at the Downing Street press conference on Monday, the minister said choosing a moment to relax restrictions was “difficult”.
Boris Johnson’s government faces calls to open schools before the Easter holiday in April.
Ministers are set to conduct a review of lockdown on February 15, Downing Street confirmed.
But Hancock underlined there were 37,000 people in hospital with coronavirus, which he said was “almost twice as many as at the first peak back in April”.
He also noted there are “more people on ventilators than at any time in this whole pandemic”.
He added: “The pressure on the NHS remains huge and we’ve got to get that case rate down. Of course I understand the yearning people have to get out of this.
“The thing is that we have to look at the facts on the ground and we have to monitor those facts.
“And of course, everybody wants to have a timeline for that, but I think most people understand why it is difficult to put a timeline on it because it’s a matter of monitoring the data, and in fact this is a state-contingent and not a time-contingent question.
“This is not a moment to ease up. The success of the vaccine rollout means we cannot put this progress at risk.”
Hancock also cast doubt over whether England would return to the four-tiered system of regional lockdowns that differed across the country.
He said ministers were holding discussions about what system would replace the lockdown once it is lifted.
“We are looking at this question right now and we are looking at the impact that the different tiers had, and we’re also looking at where we found transmission happened,” he said.
“That is a question that will be driven by the science.”
It comes as the latest figures showed 6,573,570 people in the UK had received a first dose of vaccine – a rise of 220,249 on Sunday,
Pressed by a member of the public, however, Susan Hopkins, Covid-19 strategic response director of Public Health England, said the UK was still “far away” from achieving herd immunity.
“One in 10 people having immunity is far away from where we need to be,” she said.
“Herd immunity is a word we use to describe the immunity we get from vaccination.
“The big job here is to roll out the vaccination to those individuals first of all, to those who are high risk of death and hospitalisation and then to the rest of the population.
“Once we have done that, then we will have herd immunity.”