The bank last week announced that its branch in Forfar will close its doors to the public in April next year.
The branch is one of 9 in Scotland to be axed, part of a programme to close 70 across the country.
It follows the 2017 closure of its Kirriemuir premises and those in Brechin and Montrose over the last 18 months, meaning customers must now travel to Arbroath to be seen by their bank.
Ms White has notified bosses that driving across Angus “just isn’t an option” for many residents, and has asked for a meeting with chief executive Robin Bulloch over what needs to be done to offer a satisfactory service.
The bank said that the move has been prompted by more customers carrying out transactions using online services. Plans to reduce the number of branches were set out two years ago, but the Covid-19 pandemic has accelerated the shift away from face-to-face services, with customers shopping and carrying out more transactions online.
It also said that following the closures, 220 branches will remain and that more than 90% of customers will still be able to travel to a TSB branch in 20 minutes or less.
Ms White said: “Angus is losing several high street banks every year, with some towns like Kirriemuir now without a single branch.
“One bank to serve a population of 100,000 people doesn’t serve the people of Angus well enough.
"TSB isn’t the first to do this — Santander closed everything bar the Arbroath branch — but this trend can’t continue as it negatively affects the local population – not everyone has access to a computer, Broadband or feels comfortable with internet banking.
“Banks should reconsider moving wholly online while there is still a sizeable community who rely on cash services – shops, charities, business account holders and some older people who don’t bank online. There is also a large population in the Angus Glens, for example, who have poor broadband.
"So I will put it to TSB that they need to offer services in the community, beyond online with the challenges that can create.”