Last month, an quake with a magnitude of 6.4 struck 1 km southwest of Lombok, killing 16 people and injuring many others.
By late Monday evening no Australians had been reported hurt in the quake.
National disaster agency spokesman Sutopo Purwo Nugroho told reporters some 200 "domestic and foreign" tourists had already been taken off the islands.
"There are challenges: the roads were damaged, three bridges were also damaged, some locations are hard to reach and we don't have enough personnel", said Nugroho.
A 6.4-magnitude quake struck the same region on July 29, which caused some damage, the nation's officials said. "So we had to get on a boat".
"Thank you for your concern - we're all fine - a bit shaken up", he said.
The quake on Sunday exacerbated those problems.
The main tourist areas of Lombok in the south and west of the island appear to have been spared the worst of the damage.
Shallow quakes tend to cause more damage than deeper ones.
Oxfam said it was providing clean drinking water and tarpaulin shelter sheets to 5,000 people and planned to intensify aid delivery.
A video of an Indonesian imam stoically reciting evening prayers in Bali as a deadly quake struck neighbouring Lombok has gone viral on social media, with internet users praising him for his unwavering faith.
Mataram residents described a powerful jolt that sent people fleeing from buildings.
Nugroho said more than 20,000 people had been displaced.
"We are just thankful to god and also to the hotel staff who really helped us".
Outside the hospital, the rubble-strewn streets of Mataram were empty, save for a few survivors picking nervously through the ruins. The natural disaster affected the three districts of North Lombok, East Lombok and West Lombok.
Najmul Akhyar, district chief of North Lombok, told MetroTV that there was an electrical blackout so he was unable to assess the entire situation, but that at least three people had been killed. By nightfall they were pulling out, saying other areas, including another collapsed mosque, needed their heavy equipment and workers more urgently.
As authorities scrambled to assess the extent of damage in Lombok, some tourists were trying to leave.
An Indonesian official says at least three people have been killed after a strong quake struck the popular tourist island of Lombok.
Singapore Law and Home Affairs Minister K. Shanmugam, who was in the Lombok town of Mataram at the time of the quake, wrote on Facebook that his 10th-floor hotel room shook violently and walls cracked.
"It was quite impossible to stand up". Four agencies including the military and the national search and rescue agency are involved in the evacuation.
Flights from Auckland to Bali are operating as scheduled, Auckland International Airport departures information showed.
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The Indonesian island of Lombok has been hit by a strong tremor, as the search continues for victims of Sunday's quake . He said the agency has sent 20 water trucks to five remote areas, including one village of about 1,200 households.
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Challenging conditions, including more than 120 powerful aftershocks, meant numerous hikers were unlikely to reach the bottom... On the Indonesian island of Lombok with mount Rinjani managed to evacuate 540 climbers, who were trapped after a strong quake .
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