The government says extreme rain and wind is predicted to continue for several days.
It says the heavy rain can cause flooding and landslides, and flood waters and mud can be dangerous for health, and increase the risk of disease.
It says the risk of serious diseases, including hepatitis A and typhoid, increases with reduced access to clean water and sanitation.
The Solomon Islands Health Ministry and the World Health Organisation are urging the public to take action to protect themselves and their families.
The WHO advises washing hands with clean water and soap, preparing and only eating safe food, boiling drinking water, and preventing mosquito bites to help people stay healthy.
The out-of-season category one cyclone Raquel swept through the southern part of the Solomon Islands at the end of last week, before it was downgraded to a tropical depression.
But the storm still caused widespread damage in Malaita, Western, Choiseul and Isabel provinces, destroying food gardens and more than 150 buildings.
One person died when they were struck by a falling tree, while eight people are missing at sea.
The Red Cross says a full damage assessment is yet to be done, and shelter kits need to be distributed to help those who have lost their home.