The Affordable Care Act’s third open enrollment season got under way, with a new array of health plans that show how the law’s influence is starting to transform the insurance industry.

Sunday’s kickoff appeared to go relatively smoothly, with little evidence of technical glitches at HealthCare.gov as consumers started to shop for coverage that will take effect in 2016. But this enrollment season will be a challenge as the Obama administration and insurers try to lure holdouts who haven’t previously signed up for ACA coverage,...