Officers now say that the attack happened between 8pm and 9pm on Sunday, October 3, at Princess Alice Park.
The man is described as white, aged between 20 to 35 years old, of a medium to stocky build and described as "extra broad" around the shoulders. He spoke with a Birmingham accent and was wearing a grey tracksuit and black trainers at the time.
From speaking to people in the area, officers now believed that the man has been seen in the park in the days, possibly weeks, leading up to the attack.
Anyone who lives, works or travels near to the park, is asked to check their CCTV, dash-cam and doorbell footage for any suspicious behaviour or a man matching the description.
West Midlands Police Chief Inspector Wayne Carter said: "I know the local strength of feeling about this. People are rightly shocked that this has happened on their doorstep. They’re also worried by the news that this man is suspected of hanging around in the park for the past few days or even weeks.
"We only know this because people who have heard what happened and who use the park, have told us. That’s really useful information and I’d ask that people keep telling us what they know.
“We’re working at full speed to find this man. We’ve carried out fingertip searches of the park, reviewed hours of CCTV, spoken to the survivor and people in the area and we’ve got evidence being analysed by forensic experts. We’ve also checked whether there have been reports of similar attacks. At this time, we don’t believe there have been but we won’t rule anything out.
“The survivor is now at home after hospital treatment. This is a really tough time for her and her loved ones.
“I’ve dedicated more officers to patrol the area and park until this man is arrested. This will include officers in plain clothes. I want women to feel safe and be safe. I want male violence against women to end.”
Anyone with information can call West Midlands Police on 101, quoting investigation number 20/1595275/21.