Two Chicago hospitals are taking the fight against opioid-related deaths into their emergency rooms.

Advocate Health Care Inc., one of the largest health-care organizations in Illinois, is giving registered patients treated for opioid overdoses free naloxone emergency response kits before discharging them.

The pilot project is currently operating at Advocate Christ Medical Center and Advocate Children’s Hospital, both in Oak Lawn, Ill. The company intends to expand the project to some of its other facilities as well, Kate Eller, Advocate...