The FDA wants to increase its authority over international mail facilities to combat a drug addiction epidemic that the agency’s chief said has migrated beyond opioids.

About 115 Americans die every day from an opioid overdose, a rate that has quadrupled in fewer than two decades. But this problem has extended beyond opioids like OxyContin made by Purdue Pharma LP to include loperamide, which is better known under its Johnson & Johnson brand-name product Imodium. Some patients are intentionally misusing...