Many seniors rush to sign up for Medicare as soon as they're eligible. But not everyone enrolls around the time of their 65th birthday.

If you missed your initial Medicare sign-up window, you have an opportunity to enroll right now. But your coverage won't start right away, and it may end up costing you more than expected.

It's not too late to secure health coverage

Your original Medicare enrollment window spans seven months. It begins three months before the month of your 65th birthday, and it ends three months after the month in which you turn 65.

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If you don't enroll during that time, you're not necessarily out of luck. Each year, Medicare offers a general enrollment period to sign up for coverage that runs from Jan. 1 through March 31. If you enroll during this time, your coverage won't start immediately, though. Rather, it will begin the first day of the month following your enrollment.

So, let's say you decide to sign up for Medicare on Jan. 10. In that case, you're not looking at coverage until February 1.

Now for the most part, you're not looking at such a large lag in coverage. The bigger issue, however, is that you may be on the hook for a Part B premium surcharge for failing to sign up for Medicare during your initial enrollment window.

In a nutshell, if you go 12 months without Medicare coverage upon becoming eligible for it, you'll face a 10% surcharge on your monthly premiums for life. And to be clear, that's a 10% surcharge per 12-month period without coverage upon eligibility. So the longer you delay your Medicare enrollment, the higher the penalties you might face for life.

Keep in mind that if you delay your Medicare enrollment due to being covered by a qualified group health plan through a job (either yours or a spouse's), that penalty is generally waived. And from there, you actually get a special enrollment period to sign up for Medicare, so you don't necessarily need to wait for the period between January 1 and March 31. But if you never signed up for Medicare and didn't qualify for a special enrollment period, then you may want to enroll at some point between now and the end of March.

Social Security doesn't have to be a factor

Some people delay their Medicare enrollment because they want to sign up for health benefits and Social Security at the same time. But that can be a costly mistake.

Full retirement age for Social Security falls between ages 66 and 67, depending on year of birth, while Medicare eligibility begins much sooner. But there's no obligation to sign up for Social Security upon Medicare enrollment, so the two should really be treated separately.

In fact, if you're nearing your 65th birthday and aren't covered by a group health plan, you should really consider making plans to sign up for Medicare -- even if you have no intention to start collecting Social Security for many years.