Thousands of patients were waiting for routine treatment at Southern Health in June, figures show.
The Nuffield Trust said there would not be any let up for the NHS in dealing with the backlog caused by the coronavirus pandemic and that the "broken and overstretched care system" continues to hinder staff's best efforts.
NHS England figures show 4,358 patients were waiting for non-urgent elective operations or treatment at Southern Health NHS Foundation Trust at the end of June – up from 3,996 in May, and 3,192 in June 2021.
None of those had been waiting for longer than a year.
The median waiting time from referral at an NHS Trust to treatment at Southern Health was six weeks at the end of June – down from seven weeks in May.
Nationally, 6.7 million people were waiting to start treatment at the end of June.
Jessica Morris, fellow at the Nuffield Trust health think tank, said the NHS has had success in clearing the vast majority of two-year waits, but added that this will "not provide any let up for NHS staff".
Ms Morris said: "The NHS may have overcome the first hurdle, but as new figures show the mountain to climb to clear the total waiting list keeps getting bigger.
"Staffing shortages and ongoing pressure from Covid-19, and a broken and overstretched care system, continue to slow down how quickly the NHS can work through this major backlog."
Separate figures show 1.6 million patients in England were waiting for a key diagnostic test in June – the same as in May.
At Southern Health, 1,657 patients were waiting for one of eight standard tests, such as an MRI scan, non-obstetric ultrasound or gastroscopy at this time.
Of them, 127 (8%) had been waiting for at least six weeks.
Richard Murray, chief executive of The King’s Fund think tank, said: "The intense pressure on NHS and social care services has barely featured in the Conservative party leadership race, yet the new prime minister will inherit a health and care system in a state of steady crisis.
"Ensuring patients can access the care they need will require urgent and sustained action.
"If the next prime minister fails to prioritise action to shore up health and care services, they can expect the NHS and social care to slide even deeper into crisis."
The Department of Health and Social Care said it is making progress in "busting the Covid backlogs" and has rolled out over 90 community diagnostic centres, which have delivered over 1.5 million checks.
A spokesperson added: "We have also met our target to virtually eliminate waits of over two years for elective care - the first milestone in our Elective Recovery Plan.
"The pandemic put unprecedented pressure on the NHS and we are hugely grateful to staff for their tireless work."