Maternity units where dozens of babies have died or been injured are no longer rated inadequate by a care watchdog.
Nottingham University Hospitals (NUH) NHS Trust is subject to the largest review of its kind in the UK after a series of failings.
Police are also investigating the trust’s maternity services, now rated “requires improvement”.
Bereaved families said the report was a “small step”, and added “much more needs to be done”.
Dr Jack Hawkins and his wife Sarah, who both used to work for NUH, have campaigned for safe maternity services after their daughter Harriet died due to failings in 2016.
Responding to the CQC’s findings, Dr Hawkins said: “It is progress and we need to be positive about that, but our concerns are it is really the bare minimum that the public should expect.
“Some of the outstanding issues raised are critical to safety and very basic.”
Mrs Hawkins said: “‘Requires improvement’ still doesn’t mean safe and what all the families want is safe care.”
She added: “The families welcome that there may be some improvements, but we really want to stress that it seems like it is the bare essentials.
“The trust still requires improvement and that’s not anything to be celebrated.”
The regulator has judged both hospital maternity units, and the trust’s overall maternity service, as requiring improvement but said it identified sufficient recent improvement to lift their long-standing inadequate rating.
Its latest findings come a year into a major review of the trust’s maternity failings by senior midwife Donna Ockenden, who is looking at the cases of some 1,800 affected families.
She previously led a review of maternity services at Shrewsbury and Telford NHS Trust, in which it was found that 201 babies and nine mothers could have survived with better maternity care over a 20-year period.
Her findings so far in Nottingham have triggered an inquiry by Nottinghamshire Police that will run alongside her own.
CQC Midlands deputy director of operations Greg Rielly said: “At both maternity services, we saw an improvement in the level of care being provided to people and their babies since we last rated both services as inadequate.
“It is positive to see that the trust is now on an improvement journey to bring about better and safer care.”
However, the CQC’s latest report said the trust’s maternity service “did not have enough substantive staff to care for women and keep them safe” and not all staff had training in key skills or assessed risks to women in a timely way.
Inspectors raised safety concerns about how medicines were managed and said not all equipment was cleaned between uses.
They also found storage of expressed breast milk was “unsafe”.
The trust was also “not always complying with its statutory responsibilities for duty of candour” – a professional responsibility to be honest with patients when things go wrong.
The CQC said further inspections would be carried out to ensure improvements in maternity continued.
The watchdog also assessed the NUH’s leadership, previously rated inadequate, which it said now “requires improvement”.