The number of people who had offered to help topped 170,000 early on Tuesday morning, according to GoodSAM, a group coordinating the response.
NHS Volunteer Responders aims to support vulnerable people during the coronavirus pandemic, as well as the health service as it tackles the outbreak.
Tasks include transporting healthy patients home from hospital, picking up essential supplies for those self-isolating, and giving someone a call who may be at risk of loneliness while staying at home.
Unveiling the scheme on Monday, the health secretary said he wanted to recruit 250,000 people.
Mr Hancock said: "We are seeking a quarter of a million volunteers, people in good health to help the NHS, for shopping, for the delivery of medicines and to support those who are shielding to protect their own health."
More than 11,700 recently retired NHS staff have also responded to a call to come back to work for the health service during the outbreak, according to the health secretary.
Some 5,500 final-year medics and 18,700 final-year student nurses would "move to the frontline" next week, he said.
The UK went into a lockdown earlier this week as it battles the pandemic's spread, with people ordered to stay at home unless for food, health reasons, or work (if working from home is not possible).
Additional reporting by Press Association