At least 14 people have died, and more than 160 were injured.
The 6.4-magnitude quake in Lombok, which was adjacent to Bali, has killed 17 people, including the Malaysian female tourist and a domestic tourist, Indonesia's national Disaster Mitigation Agency (BNPB) said.
Helicopters and rescue teams on foot have been deployed to scour the slopes of Mount Rinjani, which is crisscrossed with hiking routes popular with tourists.
Hikers were able to start climbing down on Monday after guides discovered an alternate route that was unaffected by the landslides.
Hundreds of other hikers managed to get off the mountain on Sunday, according to officials.
The hikers included 189 foreign tourists, 173 locals, 31 guides and 150 porters, "all healthy and safe", Sutopo said.
The rescuers are now evacuating 10 other climbers after finding them earlier on Tuesday.
Joko Widodo, the Indonesian President, visited Lombok on Monday morning to meet with various villages who have been impacted by the quake.
The volcano, which rises 3,726m (12,224ft) above sea level and is the second-highest one in Indonesia, is a favourite among sightseers.
"On behalf of the people of Singapore, I offer my heartfelt condolences to Indonesia and the families affected by the disaster", said PM Lee in his letter.
A total of 7,593 people are staying in temporary shelters, according to officials.
Hiking at Mount Rinjani had been suspended due to landslides but the Bali airport was operating normally, the U.S. Embassy in Jakarta tweeted.
The U.S. Geological Survey said the epicenter of the 6.4 magnitude quake was 0.8 miles west of Lelongken, Indonesia, and had a depth of 4.4 miles.
Indonesia, one of the most disaster-prone nations on earth, straddles the so-called Pacific "Ring of Fire", where tectonic plates collide and a large portion of the world's volcanic eruptions and earthquakes occur.