Image caption Locations offering in-person help to complete the online enrolment process saw increased demand in the past few days

The US government has offered help to anyone who missed the 24 December deadline to enrol for the new health insurance - known as "Obamacare".

It said anyone who failed to finish their application through a dedicated website - HealthCare.gov - could still obtain a policy for the new year.

On Monday, the government extended the original deadline by a day amid high demand and technical issues.

HealthCare.gov has been plagued by glitches since its 1 October rollout.

The problems with President Barack Obama's signature bill have hit his popularity ratings hard.

The website was set up under a 2010 healthcare law that seeks to cover millions of Americans who do not have health insurance.

'Don't worry'

Obamacare setbacks Nov 27: Year's delay in online insurance enrolment for small businesses

Year's delay in online insurance enrolment for small businesses Nov 26: Spanish-language sign-up tool is postponed until December

Spanish-language sign-up tool is postponed until December Nov 22: Enrolment deadline for individuals is pushed back a week in December

Enrolment deadline for individuals is pushed back a week in December Nov 14: Obama announces insurers can keep customers on existing plans for another year

Obama announces insurers can keep customers on existing plans for another year Oct 23: Deadline for individuals to avoid penalties is pushed back six weeks to March 2014

Deadline for individuals to avoid penalties is pushed back six weeks to March 2014 July: Businesses with over 50 workers given until 2015 to provide insurance or pay a penalty

"Sometimes despite your best efforts, you might have run into delays caused by heavy traffic to HealthCare.gov, maintenance periods, or other issues with our systems that prevented you from finishing the process on time," a message on the federal marketplace website said just hours before the Tuesday midnight (0500 GMT) deadline.

"If this happened to you, don't worry - we still may be able to help you get covered as soon as 1 January."

The website also provided helpline contact numbers and told applicants to "tell our customer service representative that you've been trying to enrol and explain why you couldn't finish by the deadline.

"They can tell you what you can do to finish your enrolment and still get covered for 2014."

Two states have extended the deadline for coverage in the new year even further, to 27 December in Maryland and 31 December in Minnesota.

While the website has been upgraded over the course of the past two months, it has still experienced downtime in the past few weeks.

Americans may still enrol in private insurance plans through the website after Tuesday, but they will not be guaranteed coverage at the start of the new year.

A more important deadline for the Obama administration is 31 March, when enrolment in the programme ends for the year.

Those who do not have coverage through their employer, government-run health programmes or through the federal or state-run websites by then will have to pay a tax penalty.

Last week, President Obama said that more than one million people had enrolled in private insurance through federal and state-run websites since 1 October.

Previously White House estimates called for 3.3 million Americans to enrol by the end of 2013, and about 7 million by the end of March 2014.