The prominent 'Dunedin sound' musician, who co-founded The Clean, was last seen at The Palms shopping mall in Christchurch on Sunday.
Police now say they are "following lines of enquiry" after "a number of reported sightings."
"Police are still working to locate Hamish, and encourage the public to report any sightings via 105."
Kilgour was inducted into the New Zealand Music Hall of Fame in 2017 with The Clean.
Police and Kilgour’s family and friends have concerns for his wellbeing, police said.
• Anyone with information on Kilgour’s whereabouts is asked to contact police on 105 or online and quote file number 221128/8475