The Premier of Temotu Province, Charles Brown Beu, says emergency supplies such as rice, water and clothing are beginning to reach those affected by the disaster.
Mr Beu says the relief effort is being hampered by repeated aftershocks that have rattled the remote Santa Cruz islands since a magnitude 8.0 earthquake triggered a metre-high tsunami last Wednesday.
He told Radio Australia's Pacific Beat many people sheltering in makeshift camps are without water and sanitation.
"What the people need most right now is water because the water source that supplies Lata and the villages around Lata township, that water source was destroyed by the tsunami.
"Up in the bush there is no water, people have to literally carry water from Lata and other sources around here in containers for cooking, drinking and general washing. In the near future my fear is an outbreak of diarrhoea and things like that."
The confirmed death toll has risen to 10, while more than 3,000 people are squatting in shelters after their homes were destroyed.
"Many of the people that have lost their homes are still gathered up on the hill where there was a temporary camp site set up initially by themselves with the help of the government," Mr Beu said.
"They have not been able to go back to their villages because virtually nothing is there, and of course they are still scared."
The latest aftershock on Monday morning measured 6.3 and was centred 51 kilometres southwest of Lata, at a depth of 35 kilometres, according to the US Geological Survey.
One measuring 7.1 on the Richter scale has caused significant damage to Temotu's only wharf in Lata.
Australia's High Commissioner to Solomon Islands, Matthew Anderson, told Radio Australia on Monday the aftershocks had been "quite significant".
"All of these things add up in terms of just additional challenges for the very, very hardy people that are on the ground providing much-needed assistance," he said.
Australia, NZ aid
The Solomon Islands Government has declared the Santa Cruz Islands a disaster area. Aerial surveys indicate most of the damage is confined to the Lata region.
Australia's Foreign Minister Bob Carr has visited the region, saying more assistance is needed on the ground.
Senator Carr says Australia will send two AUSAID workers to assess the damage and help coordinate the relief effort.
Australia is also funding a flight to take medical staff to the affected region and to evacuate severely injured people for treatment in capital Honiara.
Senator Carr says Australia is standing by to help the Solomons in the recovery process.
"The people of the Solomons are a very resilient, hardy people," he said.
"Their prime minister is determined to see that they rebuild quickly after this disaster and they've got Australia as a proven and good friend of the Solomons standing by and lending a hand."
A New Zealand Air Force Boeing 757 departed for Honiara on Monday carrying sanitation kits, tarpaulins, water containers and chainsaws packs.
The Solomons are part of the "Ring of Fire", a zone of tectonic activity around the Pacific that is subject to frequent earthquakes and volcanic eruptions.
In 2007 a tsunami following a magnitude 8.0 earthquake killed at least 52 people in the Solomons and left thousands homeless.