If there's any chance that your trip might extend beyond the period of your insurance, make sure that you know in advance how you can extend the policy and whether you can do this while traveling. It is generally much easier to extend a policy if you request the extension while you're still covered: obtaining a policy when you're traveling but aren't presently insured is difficult. In addition, if you let your policy lapse you will obviously not be covered for anything, including medical expenses, while you arrange a new policy. Many policies require that you apply for an extension at least 7 days before your policy expires.
When planning a trip, pay for your policy to cover a few days after your intended return. In the event of last-minute delays or changes of plan which extend your trip, you then have a few days to sort out any extension of the policy you need. Some policies will automatically extend if the delay is part of a problem that you could claim for: for example, if you have insured against delays and the delay extends your travel past the end of your coverage, the policy is automatically extended. This will typically not apply to pre-existing conditions or to high risk travelers such as the elderly, even if you fully disclosed pre-existing conditions when applying for cover.
Extensions to your policy are never guaranteed to happen. They will always be at the discretion of the insurer and may be refused based on your previous claims and any other information that you disclose when applying for the extension (and you will usually be required to disclose anything you think might be relevant lest the policy be void). Some insurers treat all applications for an extension as an entirely new application and will re-evaluate your circumstances before insuring you. Medical problems that occur during your holiday may count as pre-existing conditions when applying for an extension and certainly will if applying for a second policy.