Freda Walker was killed and her husband Ken, 88, left fighting for his life after being attacked in Langwith Junction, on Saturday morning.
Mr Walker, a retired miner and former chairman of Bolsover District Council, remains in hospital with life-threatening injuries in a critical condition.
No arrests have been made and residents in the village have been warned to keep their doors and windows locked while the killer remains on the run.
Derbyshire Police revealed on Monday they were keeping an "open mind" about whether the attack was linked to a violent burglary in Trowell, Notts.
During that incident, a robber broke in while a couple, in their 80s, slept and demanded money as he repeatedly punched them on January 6.
Today the force said it was also looking at a number of similar offences that may have happened "months, years or weeks" ago to see if they are connected.
Assistant Chief Constable Dave Kirby, of Derbyshire Police, branded the murder "an act of horrendous cowardice" as he re-appealed for help from the public.
Speaking at the scene, he said: "Ken remains in a critical condition in hospital.
"Our thoughts are with him and friends and family and we wish him all the best but he is still in a critical condition.
"We are following a number of significant lines of enquiry, lots of work to do and we have a huge amount of dedicated resources deployed.
"There will be somebody out there in the community who will know something that might help us.
"That might be something they have seen that was out of the ordinary or a piece of information they think might be useful.
"It might seem tiny and insignificant but quite often those little pieces of information are what can take an investigation in a different direction.
"My appeal very strongly is that if anybody has any information of that type please ,without hesitation, come to us.
"We can treat that information in confidence and please do that as a matter of urgency.
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"Our investigation will find out who is responsible but we want to do that as quickly as we can.
"Ultimately, somebody is responsible for this terrible tragedy, an act of horrendous cowardice, and they are currently loose.
"They are currently out in the community. I'm not suggesting that people are now in danger or under attack but I am saying we need to get hold of those people straight away.
"Nowadays there are so many cameras around, public and privately owned, and there is a lot of work going into what is on those cameras and analysing that.
"There are also a number of enquiries which are a bit more sensitive, I wouldn't chose to talk about.
"But we are looking at a number of similar offences that may have happened months, years or weeks beforehand and looking to see if we can draw similarities and bring the investigations together ultimately to solve all of those crimes.
"The incident in Trowell is being led by Notts Police, the two investigation teams are working closely together to ensure all information is shared.
"At the moment we are not formally linking those albeit some of the way the crimes took place are similar.
"It is tragic this could happen to anybody but doubly so that Ken and Freda are, and were, a real backbone here.
"They have done so much work for the good of other people and that's what their lives were about.
"They were married for 60 years and all of the time lived in this house. Freda was fit and well up until this incident even at 86-years-old.
"She was a member of local clubs including a knitting club, her whole career was in the textiles industry. Ken was a pit man, a staunch Labour supporter.
"He was a member of the council for 21 years and gave so much to that, serving actively on many committees, raising money, they both did, for local charities to make life better for people.
"He was offered an Honoury Alderman title at the end of last year. So for this to happen to a couple like that, who have given so much, is doubly tragic."
Shocked residents previously paid tribute to Mrs Walker, describing her as a "generous person with a warm spirit" who was known for her charity work.
One neighbour said: “Freda was the most generous person you could ever meet andhad such a kind and warm spirit.
"The couple didn't have any children and I think helping others was her way of being a mother to those who were less fortunate.
“She worked with homeless charities and just wanted people to have a chance to make a life for themselves.
“The community here are in real shock. It’s so sad and an awful thing to happen to two pillars of their community.”
A friend of Mr Walker added: "Everyone's numb with shock. Ken is a lovely man and was devoted to Freda.
"They didn't have children but gave all their time to helping others. Ken was still very active on the council and had no plans to retire.
"Freda was dedicated to her charity work and was passionate about helping rough sleepers."