On Thursday morning, Mr Hancock said the number of people who have already volunteered their time was "fantastic" and invited more to join the effort.
Stuart Beards, 26, hopes to help by delivering necessities to those in quarantine, saying he was motivated by "the countless pleas online from health officials to support the NHS".
He said: "It needs to be a universal community effort to support the NHS."
When asked about any concerns for his own health, the London-based security operations manager said he was "already taking precautions at home and are sticking to the rules implemented by the Government and health advice to minimise any contact".
He added: "At the end of the day, if we can help by volunteering, it may stop a vulnerable person leaving their home.
"If it helps keep one person safe, it's worth it."
Lisa O'Hare, 44, from Wilmslow, has also signed up and wants to give back to the health service that has helped her.
"I've got one dodgy kidney but I wouldn't like the other to be fighting on its own," she said.
"If it wasn't for the NHS, I wouldn't be here now. They've saved me and so many people I love so many times. This is the least I can do."
The tax manager will be making calls to vulnerable people as she has an underlying health condition and does not wish to risk exposing herself to the virus.
She added: "It felt like something I could do from my house which is useful - a lot of people could do with a chat and I could too, so it will be useful."
Owen Lloyd, 22, volunteered after seeing the strain his NHS staff friends are under.
He lives a few minutes' walk away from east London's ExCeL Centre, currently being transformed into NHS Nightingale, and said: "I'd like to offer any support when that is ready."
Mr Lloyd said: "Even if I only end up doing one thing through this, I will be proud to say I have assisted the NHS and taken some pressure off our brilliant NHS workers."
Jacqui Hargrave, 48, usually writes a popular blog about open water swimming but work is restricted as her family needs to stay home, so she will "be making phone calls to people who are lonely".
She said: "My parents, they are in their 70s and vulnerable but they have each other.
"They have neighbours who have nobody, and they will be so isolated, particularly the elderly.
"If I can help even one person so they can have something to look forward to during the day then it's a good thing."
The Prime Minister praised those who have volunteered during his daily press conference on Wednesday.
Speaking from Number 10, Boris Johnson offered the then-405,000 who had signed up a "special thank you".
He added: "When we launched the appeal last night, we hoped to get 250,000 volunteers over a few days" and said the volunteers would be "absolutely crucial" in the fight against the spread of the illness.