Mr Hancock himself warned Britons not to book a summer holiday abroad just a few weeks ago.
Since then more than 8.4 million people have received at least one Covid jab.
The government says it is also on track to vaccinate everyone in English care homes by the end of today.
During an appearance on the BBC’s Politics East programme Mr Hancock said: “In six months we’ll be in the middle, I hope, of a happy and free Great British summer – I have a high degree of confidence that by then the vast majority of adults will have been vaccinated.”
He added: “I think we are going to have a great summer, but we will have a tough few months between now and then.”
And ministers “still don’t know” when they would be able to lift restrictions imposed as part of the latest lockdown.
The government has been warned that social distancing rules may have to stay in place unless a vaccine proves to be 85 per cent effective at stopping transmission of the virus, as well as at preventing severe illness.
The warning comes from SPI-M, a subgroup of the Sage group of government advisers.
Asked about the modelling, Ms Truss said she did not “want to make predictions about the situation in the autumn, I think it’s far too far away”.
She added: “Long-term predictions in what is a very, very unpredictable situation are not wise.”
On Saturday, the World Health Organisation urged the UK to pause its vaccination scheme once vulnerable groups had been inoculated, in order to contribute to a global rollout.
Ms Truss said it was “too early” to talk about when the UK might give excess doses to other countries.
Meanwhile, Public Health England’s Dr Susan Hopkins warned that relaxing lockdown measures would have to be done “very slowly, very cautiously” to avoid a surge in infections.
“We have learnt, as we did on the first occasion, we have to relax things really quite slowly, so that if cases start to increase we can clamp down quite fast,” she told BBC One’s Andrew Marr Show.
“The NHS is going to be under pressure until the end of March, as normal in winter, but even more so with the amount of inpatients they still have with Covid-19.
“Any releases that we have will have to happen very slowly, very cautiously, watching and waiting as we go, with a two-week period to watch and see the impact of that relaxation because it takes that to see what’s happening in the population.”