The first batch of figures for the service showed that out of 8,117 people whose details it received between May 28 and June 3, 5,407 were subsequently contacted.
Baroness Dido Harding, who is overseeing the programme, told Thursday’s Downing Street briefing: “I think for the first week of a scale citizen service this is good performance.
“And clearly it can and needs to and will get better.”
From the 5,407 people with Covid-19 that were contacted, a total of 31,794 close contacts were subsequently identified as being at potential risk of having also been infected.
Of these, 26,985, or 85 per cent – were then reached and asked to self-isolate – the vast majority within 24 hours.
But the service was unable to reach 4,809 people, 15 per cent of the total “at risk” group.
Baroness Harding, chairwoman of NHS Improvement and former chief executive of the TalkTalk Group, said that NHS tracers to try to contact people ten times during a 24-hour period.
She added: “If you look at why we haven’t got 100 per cent of people’s close contacts, some of it is we all have a part to play in this which is when take a test making sure that we give the right contact details for ourselves so it’s easier for the Test and Trace team to contact you.
“Some of it is us learning when is it most convenient to reach people.”
She added: “Can we do better? Yes of course we can and we will continue to work at it.”
Health Secretary Matt Hancock said the system had beaten his expectations.
He told the press conference: “I think that the system has worked well and to get two-thirds in the first week of operation, it beat my expectations.
“Then to have the vast majority – 85 per cen – of the contacts that were given self-isolating, that beat my expectations too.”
The minister also said the Government had not ruled out the introduction of an enforcement mechanism if people failed to comply with the test and trace system.
“We’re not ruling it out at the moment but we don’t think we need it,” he told the briefing.
Baroness Harding said the “vast majority” of people were complying with the programme.
She added however: “Making sure that we bring as many people as possible who have got coronavirus into the test and trace programme is the number one thing we need to improve over the coming weeks.”