The NHS has hailed reaching the "home straight" of the vaccination rollout, with people aged 25 and over now being invited to come forward.
Three-quarters of the adult population across England have had their first dose, while more than half are fully vaccinated, six months on from the country's first jab.
But figures suggest there is still hesitancy amongst over-50s in some areas, particularly in London and the major cities, which could slow the race against the virus as the dominant "Indian variant", now known as the Delta variant, claims more cases.
People aged 50 and over have been invited to book their first jab through the NHS since March 17.
NHS England data shows in Shropshire, 6,409 people aged 50 and over had still to receive a first vaccination dose as of May 30 – four per cent of the age group based on latest population estimates from the Office for National Statistics.
The figures for the Telford & Wrekin Council area show that virtually all over 50s have had their first jab.
Across England, four per cent of people aged 50 and over – 800,600 adults – had not been vaccinated as of May 30.
Regionally, London had the lowest uptake with 11 per cent of people aged 50 and over unvaccinated, while the South West had the highest, with just two per cent unvaccinated.
In the Midlands, three per cent of people aged 50 and over were unvaccinated.
Last month, the NHS released separate data on ethnicity showing vaccine uptake across England among black Caribbean people was the lowest, with 37 per cent of those aged 50 and over unvaccinated by May 7.
The group with the highest uptake was white British, with five per cent unvaccinated.
The data also showed nine per cent of people aged 50 and over in the most deprived areas were unvaccinated, compared to two per cent in the most affluent.
Dr Perpetua Emeagi, a lecturer in human biology and biological sciences at Liverpool Hope University, said not enough information had been given to ease a deep scepticism surrounding the production of vaccines among the black, Asian and minority ethnic community.
She said: “Materials on the Government website are perhaps tricky to find and are aimed predominantly at health professionals rather than the average person.
“We need to do more to get this information out to those with real concerns about a vaccination programme they’re reluctant to embrace."
Labour's shadow women and equalities secretary, Marsha de Cordova, said the vaccination programme had "exacerbated existing health inequalities of race and ethnicity".
She added: "These next few weeks are crucial in containing outbreaks of the Delta variant - increasing vaccine uptake must be ministers' number one priority.”
Despite this, NHS England said vaccine uptake in black British and Asian communities tripled between February and April.
NHS national medical director for primary care, Dr Nikki Kanani, said this was achieved through talks with faith leaders, pop-up clinics in places of worship and community centres, and a strong backing from celebrities.