A tsunami warning was in effect in the Solomon Islands, Vanuatu and Papua New Guinea, although it was not known whether a tsunami had been generated.
The latest quake was registered at a depth of 12 miles below the seabed, the US Geological Survey said.
There were no reports of major damage or casualties after the first quake early Sunday local time.
The first tremor woke people in the Solomon Islands, striking at 07:14 local time, and was followed by a 5.9 magnitude aftershock.
The Pacific Tsunami Warning Centre reported indications of a small tsunami which may have caused some damage hear the epicenter 200 miles southeast of the capital, Honiara.
The Solomon Islands lies in a zone prone to frequent earthquakes and volcanic activity.
In February, several people were killed after a tsunami triggered by a magnitude 8.0 quake struck near the Santa Cruz islands – part of the Solomon Islands nation.
The latest quakes come only a week after the Solomon Islands were hit by severe flooding, which left more than 20 people dead.
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