Mr Chalerm said Friday Prime Minister Yingluck Shinawatra had assigned him to oversee the safety of Thai and foreign tourists after Mrs Smith, 60, was stabbed to death on Wednesday night.
Mrs Smith and her friend Tammee Lynn, also an Australian national, were walking back to their hotel in the Kata Noi area at around 10.30pm when they were attacked by bag-snatchers. Mrs Smith died instantly from stab wounds while Ms Lynn was wounded.
Mr Chalerm said he had told Pol Lt Gen Santi Pensuit, commissioner of Provincial Police Region 8, which covers the southern provinces, to apprehend the two suspects as quickly as possible.
National police chief Gen Priewpan Damapong had also ordered his deputy, Pol Gen Pansiri Prapawat, to oversee the case in the area, Mr Chalerm said.
He adding that the government regarded this case as "a significant issue".
Phuket police's deputy commissioner Pol Col Peerayuth Karachedi said four witnesses have already been questioned.
The licence plate of the motorcycle on which the attackers escaped was captured on CCTV footage and that line of inquiry would be investigated, he said.
According to the preliminary probe, the killers are likely to be members of a motorcycle gang known for committing robberies around Phuket.
Phuket deputy governor Chamroen Thippongthada said the province would send letters of condolence to the families of the victim and the injured woman, the Post reported.
Authorities will be called to an urgent meeting to discuss the incident and any preventive measures that could be implemented to prevent repeat attacks, Mr Chamroen said.
Australia's honorary consul Larry Cunningham visited Ms Lee and viewed the body of the murdered woman.
He was ready to cooperate with police investigators and had contacted the victim's family in Perth, Mr Cunningham was quoted as saying.
Tourism Minister Chumpol Silpa-archa said after meeting Mr Chalerm that the incident has damaged tourists' confidence in Thailand, as well as the country's overseas image.
The Ministry of Tourism and Sports and the Tourist Police Division will devise measures to improve tourist safety, Mr Chumpol was quoted by the Bangkok Post as saying.
Phuket Tourist Association president Somboon Chirayus said the incident had had a tremendous impact on the province's image and directly affected tourism there. Many overseas travel agents had asked him about the attack.
Mr Somboon said Australian tourists are an important market, but so far no tour group had been cancelled. Current accommodation bookings remain at 50 per cent of availability.
He also said he wanted to see more police assigned to provide safety for tourists in Phuket and more CCTV cameras to be installed in some areas.