The Captain Tom Foundation, which was set up to support causes close to Sir Tom’s heart, said its work would “aspire to ensure Tom’s message of hope becomes an enduring legacy”.
Singer Michael Ball, who recorded a charity single with Sir Tom which reached number one, said the Second World War veteran had left the public richer through his charitable acts.
Speaking on BBC Breakfast, he said it was “an awful irony that he has been taken by this virus that he fought so hard to tackle”.
But he added: “He’s left us richer, hasn’t he, in every sense, and better off. He set an example for us all.”
Sir Tom had been taken to hospital on Sunday after being treated for pneumonia for some time and testing positive for coronavirus last week.
His family praised the care he had received from the NHS and said they had been able to spend time with him in his final hours.
In a statement, his daughters, Hannah Ingram-Moore and Lucy Teixeira, said: “We are so grateful that we were with him during the last hours of his life; Hannah, Benjie and Georgia by his bedside and Lucy on FaceTime.
“We spent hours chatting to him, reminiscing about our childhood and our wonderful mother. We shared laughter and tears together.”
They added: “Whilst he’d been in so many hearts for just a short time, he was an incredible father and grandfather, and he will stay alive in our hearts forever.”
Prime Minister Boris Johnson described Sir Tom as “not just a national inspiration but a beacon of hope for the world”.
“In the face of this country’s deepest post-war crisis, he united us all, he cheered us all up, and he embodied the triumph of the human spirit,” he said.
Buckingham Palace said the Queen would be sending a private message of condolence to Sir Tom’s family, while the White House also joined the chorus of tributes.
Former England cricket captain Michael Vaughan said Sir Tom’s legacy will inspire people to be better.
Vaughan, who awarded the staunch cricket fan honorary membership of the national side on his 100th birthday last year, told BBC Breakfast people “can take a huge lesson from Sir Tom about being a better person”.
Sir Tom’s fundraising efforts went on to inspire others to take on similar charity challenges, including Dabirul Islam Choudhury, who raised more than £420,000 by walking 970 laps of his communal garden in London while fasting during the Islamic holy month of Ramadan.
Mr Choudhury said: “He is our real hero and we will miss him terribly.”
The flag above 10 Downing Street was flown at half-mast on Tuesday evening, while landmarks including the London Eye, Wembley Stadium and Blackpool Tower were lit up and a tribute was broadcast on the billboard lights at Piccadilly Circus.
Floral tributes were laid at the gate to his family home in the village of Marston Moretaine in Bedfordshire, as neighbours described him as “a legend, inspiring, a hero”.