AMC Entertainment Holdings Inc.’s movie-subscription program has grown faster than the theater chain expected, giving the company a predictable revenue stream as it battles other movie-theater operators and streaming services like Netflix Inc. for consumer attention.

The company said Wednesday more than 100,000 people signed up for the AMC Stubs A-List program over the past six weeks, sending the service’s subscriber count past 600,000 since it launched in late June. Company officials had set a target for the service to have...