Dept Ch Const Critchley said: "I am still appealing for information in relation to anybody who knows who the person was responsible for pulling that trigger in the most indiscriminate, most appalling way, anybody who knows where the weapon is, anybody who has harboured or has any information at all. We still want to hear from them."
He added investigators were working around the clock to work out what happened before, during and after the attack.
The policeman said: "We have made great strides forward but I am appealing for further information."
His comments came after Merseyside Police Chief Constable Serena Kennedy vowed the force would be "relentless" in the hunt to bring the killer to justice.
She said: "Detectives are continuing their investigations and they will be relentless in their work to establish what led up to the incident on Christmas Eve and who was responsible for this sickening and heart-breaking incident which has ruined so many lives and culminated in the loss of a beautiful young woman."
Ms Edwards' death sent shockwaves across the community where she lived. Tributes including cards, flowere and photographs have been left outside the pub where she had been celebrating Christmas with friends on the night of the fatal shooting.
The Catholic Church of the Holy Apostles and Martyrs in Wallasey is to open its doors to offer people a place to pray for Ms Edwards and her family, according to the Liverpool Echo.
Parish priest Canon Philip Moor, whose niece and nephew were in the Lighthouse at the time of the shooting, has organised the event for Saturday, December 31.
Canon Moor told the publication: "As we were celebrating the joy of the birth of Jesus, we were extremely shocked and saddened to hear about the tragic shooting on Christmas Eve.
"The local community has been devastated by this terrible event. The Lighthouse pub is right at the heart of our community and my own niece and nephew were there, along with so many others at the time of the incident. The Lighthouse for many years has been the place to gather for our young people.
Wirral Borough Council leader, Jeanette Williams, described the shooting as "horrendous and abhorrent". She said it was a "one-off occurrence which is out of character for the area".
Ms Williams added: "Our hearts go out to the family of Elle at this tragic time as they deal with the loss of their loved one.
"If you have any information, then do the right thing and speak up. No one can sit in silence."
Local councillor Lesley Rennie said residents felt mortified by what happened.
Police have said previously that officers have stepped up their presence in Birkenhead, Woodchurch and Wallasey in a direct response to the fatal shooting.
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