Shamsudden Mahmood, 26, was shot dead on 2 June as he served a customer at the crowded Mumutaz Restaurant, the island's only Indian restaurant.
His attacker, wearing a balaclava, walked into the building through the front door, shot Mr Mahmood in the head at point-blank range, then fled.
Police have only been able to speculate about the motive for the killing, or whether the killer was an islander or an outsider. They are not certain whether the killer is still on the island.
An early theory was that the victim had been selected as the target for a racial attack. However, the killer's face was so comprehensively masked that officers are not certain whether he is white.
Mr Mahmood first arrived on the island about a year ago. He is believed to have previously lived in Southampton. During his time in the islands, he lived above the restaurant with the owner's family.
A team of 20 detectives was flown in from Inverness to find the killer, and a watch is being kept on exits from the island in case the killer is still there.
A police spokesman said: 'There is no question over our resolve to catch this man. We intend now to spread our door-to-door inquiries from the immediate vicinity of the killing to the wider area.'Reuse content