WASHINGTON — President Obama’s push to cover America’s uninsured faces another big test Monday.

This time, it’s not only how the website functions, but how well the program itself works for the millions who are starting to count on it.

Midnight Monday, Pacific time, is the deadline for new customers to pick a health plan that will take effect Jan. 1, and for current enrollees to make changes that could reduce premium increases ahead of the new year.

HealthCare.gov and state insurance websites are preparing for heavy online traffic before the deadline, which gives consumers in the East three hours into Tuesday to enroll.

Wait times at the federal call center started creeping up around the middle of last week, mainly owing to a surge of current customers with questions about their coverage for next year. Many will face higher premiums, although they could ease the hit by shopping online for a better deal. Counselors reported hold times of 20 minutes or longer for the telephone help line.

About 6.7 million people now have coverage through Obama’s signature law, which offers subsidized private insurance. The administration wants to increase that to 9.1 million in 2015. To do that, the program will have to keep most of its current enrollees while signing up more than 2 million new paying customers.

People no longer can be turned down because of health problems, but picking insurance still is daunting for many consumers. They also have to navigate the process of applying for or updating federal subsidies, which can be complex for certain people, including immigrants. Many returning customers are contending with premium increases generally in the mid-to-high single digits, much more in some cases.