His partner, Alvina Smith, was one of the last people to see him at their home before he disappeared.
Police are following three more potential sightings of Edwards In Auckland, Maketu and Waikeremoana, Detective Sergeant David Whitecliffe-Davies said yesterday.
Members of the public had come forward and said they had seen someone matching Edwards' description who seemed "a little bit out of place".
Whitecliffe-Davies said there were no further updates on the two potential sightings in Whakatāne and Rotorua, which police received tips about on Thursday.
Smith said she and Edwards have been together for 18 years and have four sons aged 4, 6, 7 and 16. Smith, 43, described him as "the best dad in the world".
"Then I really started to panic. Tokoroa is such a small town ... I've lived there most of my life and Tokoroa people love to talk about each other or anything.
"And there was just no bicker - there was nothing. Everywhere we went to or everywhere I thought he may be, there [was] no sign."
In the week leading up to his disappearance, Smith said there was "nothing different" about her partner. He had helped out with the children and caught up with some friends.
Smith said his disappearance would not be gang- or drug-related.
At the time he went missing, Edwards was not working, because he was looking for a job that worked around the kids' schedules.
Smith described Edwards as kind and generous and she said he "didn't like to be away from the kids" for too long.
Her message to Edwards was: "We love him and we just want him home. We miss him."
Edwards was wearing a brown Swanndri jersey with a tinge of green, light blue jeans, black Converse Chuck Taylor shoes and a black beanie when he went missing.
He has been described as lean and between 1.80m and 1.85m tall, with dark, short, spiky hair, brown eyes, olive to white-coloured skin and no piercings or tattoos.
Whitecliffe-Davies said: "Even if you think it's a long shot, we'd like to know because we'd hate for it to be that little bit of information that we sort of miss."
Anyone with information is asked to contact Detective Sergeant Dave Whitecliffe-Davies on 021 191 0875 or police on 105 and reference police file number 220518/6338.
Alternatively, information can be provided anonymously to CrimeStoppers on 0800 555 111 or through their website.