Government and international rescuers have been struggling to reach the hardest hit areas in the remote west of the country where officials believe hundreds of homes have been destroyed.
"We are under-resourced, we need bigger vehicles," said Jonathan Taisia, a Red Cross disaster official in Gizo town.
He said they needed more workers to clear the debris which has cut off roads to outlying areas.
"I think the thing right now is water, water and tents. People are still up in the bush and are reluctant to go back to the villages"
Charles Kelly, Solomons Red Cross
Most aid was being delivered to Munda, on a nearby island, but a shortage of boats has hampered distribution efforts.
The bulk of the local fleet of canoes and other vessels were destroyed by huge waves up to 5 meters high.
The government has said thousands more stranded in remote villages and hilltop shelters may not get help for another two days due to impassable roads.
Most fear returning to their homes as strong aftershocks continue to shake the South Pacific islands.
George Herming, a government spokesman, said 5,400 homeless people may have to wait until Friday for urgent supplies of food, water and shelter to arrive.
Officials say the death toll could rise with an increasing risk of disease, including malaria.
"I think the thing right now is water, water and tents. People are still up in the bush and are reluctant to go back to the villages," said Charles Kelly, secretary general of the Solomon Islands Red Cross.
Some coastal areas are still inundated, with aerial patrols reporting seeing floating bodies.
The government has declared an emergency in the affected areas. The Red Cross has urged the homeless to return to their villages but most people are staying put fearing more tremors and tsunamis.
The country's closest neighbours, Australia and New Zealand, have offered $2 million in aid and sent around 100 soldiers who arrived on Wednesday with water, food rations, shelter and other relief supplies.
Military rescuers will also ferry essential items and supplies to the worst-hit provinces by helicopter and boat.
The tsunami caused widespread damage to homes constructed from traditional materials, while the earthquake caused more damage to sturdily-constructed buildings.
The double disaster forced the closure of food markets, disrupted fishing activities and halted domestic farming.
The islanders live scattered through a string of small islands along the Pacific's "Ring of Fire", where volcanic activity and earthquakes are common.