Mother-of-two Jean Campbell, 53, was last seen heading towards Cranhill Park while walking her dog in the east end of Glasgow on Friday December 13.
Her body was discovered at around 7.40am the next day by her husband who had gone out to search for her.
An appeal for information about her murder is appearing in the newsfeed of up to 200,000 Facebook users in the Glasgow area from today.
Detective Chief Inspector Colin Carey of the Police Scotland Major Investigation Team, said: "It has now been a month since the death of Mrs Campbell. Christmas and new year was a very difficult time for her family and friends. As you can imagine, they are devastated by their loss.
"Whilst the public response to our appeals for information has been very positive, unfortunately at this time, who killed her and the reason why is still unknown.
"Social media is an excellent way to reach thousands of people who can sometimes be hard to reach so we are keen to exploit this opportunity to obtain more information regarding her murder."
A £5,000 reward has been offered by Crimestoppers for information which leads to the arrest and conviction of whoever killed Mrs Campbell.
Mr Carey said the killer might have sustained minor injuries to their face or body during the attack and their clothing could have become dirty or stained with mud or grass.
He said: "If you know anyone who came home late on the evening of Friday 13 December or early hours of Saturday 14 December with unexplained injuries, dirt-stained clothing or was acting strangely, then please contact police.
"I really do believe that there is someone out there who is either harbouring the person responsible or knows who is responsible but has not come forward to police. This can of course be for a number of reasons, fear or intimidation being the most obvious.
"If you have information but do not wish to approach the police directly, then please call Crimestoppers, who are offering a reward of up to £5,000, where you do not have to give any personal details. We also have a dedicated email address that you can contact without saying who you are."
More than 1,100 DNA samples have been taken from men and women in the area and more than 550 homes have been visited during house-to-house inquiries.
Mrs Campbell was captured on CCTV at around 10.30pm on December 13 in the Bellrock Street area, heading towards the park.
She was wearing pyjama bottoms and a black, three-quarter length jacket with a hood.
Her German shepherd dog Kai was found in the park.