The House will finish a two-weeks-long push to pass dozens of opioid bills June 22, but addiction groups warn lawmakers may have missed their mark in combating an epidemic that killed roughly 64,000 people in the U.S. last year.

The more than 60 bills that cleared the House tackle the opioid epidemic through a wide range of changes, such as establishing grant programs and redirecting federal funds to aid drug treatment services, which could help thousands of Americans addicted to...