The number of people donating organs after death has risen 50% since 2008.
More than 1,200 people in the UK donated their organs in the last year, leading to about 3,100 transplants.
The increase has been largely credited to the network of specialist nurses who approach and support bereaved relatives in hospitals.
But with the numbers on the organ donation register remaining unchanged, the NHS Blood and Transplant service is still asking people to sign up.
Sally Johnson, from NHS Blood and Transplant, said: "The NHS has worked hard to ensure every potential donor is identified, that the organ donation register is checked, and that families are approached.
"But the NHS can't do it on its own. We need a transformation in donor and family consent because the UK's family refusal rate remains one of the highest in Europe."
Last year, 125 families overruled an individual's intention to donate.
Ms Johnson said: "It is important families know their relative is signed up to the register. On the worst day of their lives, don't let families guess"
In 2008, the UK's health ministers accepted the recommendations of the Organ Donation Taskforce.
These included better identification of potential donors, improved referral of potential donors and improved care of donors to increase the number of organs available - around three organs were retrieved last year for every donor.
Each donor has the potential to help nine people through donation of a heart, lungs, two kidneys, pancreas, liver and small bowel and two corneas.
Since the Taskforce report recommendations five years ago, Scotland and Northern Ireland have seen the largest increases in deceased donors- 74% and 82% respectively.
And the actual number of transplants in the UK is up by 30% - the difference between the increase in donors and transplants is partly down to organs from some who wish to donate being unsuitable because of factors such as age and obesity.
But with three people in the UK dying a day because of a lack of donor organs, many experts believe more need to be done.
Dr Vivienne Nathanson from the British Medical Association says the donation system needs an overhaul. Referring to recent research she said: "[We] concluded the best way forward for the UK was to introduce an opt-out system with safeguards."
The plan would mean people were deemed to consent to organ donation, unless they object during their lifetime. This system is being considered for introduction in Wales in two years time
Ministers in Northern Ireland recently announced the intention to consult on public attitudes towards organ donation, including the introduction of an opt-out system.