The U.S. Geological Survey said an earthquake measuring 6.1-magnitude rumbled off the Santa Cruz islands, following an 8-magnitude quake that struck the same region Wednesday.
Damage at a key airport in the area is preventing aircraft from landing and hindering aid efforts, officials said.
Thursday's earthquake, at a depth of about 6.2 miles, occurred off the Santa Cruz islands, about 195 miles east-southeast of Kira Kira and about 344 miles east-southeast of Honiara in the Solomons.
The three-foot wave surge inundated Lata township, dumping debris on its airstrip and forcing aid to be sent to the affected areas by boat, officials told Radio New Zealand.
Officials said they feared the death toll could rise as reports from the isolated region become available.
World Vision estimated 3,500 people were displaced by the tsunami. The organization also said some houses in the town of Venga were moved nearly 3 feet by the surge and nearly all the homes in the village of Nela were washed away.
The Solomon Times Online reported several people, including a child, were unaccounted-for and hundreds of houses have been destroyed.
A tsunami warning for New Zealand, Australia and other Pacific island nations was issued then lifted.