Jomaa Jerrare, a 52-year-old mother of one, was named by detectives on Thursday as the woman found dead in a layby next to the A454 Bridgnorth Road in the early hours of Monday.
Officers have pleaded with anyone who has seen Ms Jerrare in recent months to speak to them as they searched the high-rise building where she lived in Heath Town for evidence.
Detectives from Staffordshire Police were seen entering Hampton View on Woden Road on Thursday morning as officers from West Midlands Police kept the scene.
A spokesman for the West Midlands force confirmed the 20-storey block of flats was Ms Jerrare's last known address.
Meanwhile the stretch of Bridgnorth Road in Perton where Ms Jerrare was found, which is around four miles away from her home, is expected to remain closed for several more days as specialist officers comb the nearby fields and hedges.
Ms Jerrare is believed to have been driven to the layby before being set on fire, but it is not yet known where she was killed.
A press conference was held at the scene on Thursday afternoon, where the road is covered in cuttings from hedges which have been stripped back to help the search teams.
Detective Superintendent Tom Chisholm revealed that Ms Jerrare lived alone in Heath Town but had not been seen by her family for six weeks.
Det Supt Chisholm said she had lived a "chaotic" lifestyle involving substance abuse and it was important to find out who she had been with over the last few weeks.
The cause of Ms Jerrare's death has not been established despite a post mortem having taken place.
Initially the body was believed to belong to a woman aged between her late teens and 30 and her ethnicity was not clear due to the severity of her burns.
Police are keen to hear from anybody who was in the Bridgnorth Road area in the early hours of Monday morning, as well as anyone who may have come into contact with Ms Jerrare in recent months.
A man arrested in Finchfield after being seen acting suspiciously near the scene has been released and eliminated from the inquiry after providing an account for his movements.
Anyone with information is urged to contact Staffordshire Police or Crimestoppers.