The link between extra pounds and severe Covid-19 grew stronger as the U.S. Centers for Disease Control and Prevention said that people who are merely overweight, not just the obese, may be at high risk of serious disease from the infection.

The warning, posted on the agency’s website Tuesday, means about two-thirds of Americans could face higher risks.

Nearly 40% of American adults are obese, which the CDC tied to poor outcomes of the infection in late June, and about 32% are overweight, according to the agency. Obese people are more likely to fall very ill with Covid-19 and be hospitalized, and the risk of death from infection increases along with higher body mass index, the CDC has said.