Sure enough it depends on you plans, what would be the right solution. How long will you travel? What countries?
Stay with your local health insurance
German statutory health insurances are limited to countries with a social security agreement, i.e. Europe and a few additional countries. You should talk to the insurance company about special conditions for those countries, what documents you need to take with you and how long you are covered. According to my experience (Spain) it is kind of a drama to get foreign doctors to accept the german insurance card. They usually try to make you pay in advance. On top there are a lot of private hospitals, that wouldn't accept it at all. Nevertheless it is always possible to pay the costs on the spot and submit the invoice to your health insurance. Just be aware, that it might not be refunded or you just get a partial refund depending on the insurance terms.
German private health insurances usually cover all countries but just for a certain time. Mine is limited to 6 months and it still does not cover ALL of the medical expenses that might occur in a foreign country. I talked to my insurance company and they gave me the advice to set my insurance on hold (re-entry without medical examination) and get an overseas health insurance instead. Would be cheaper and those would be tailor-made for travel incidents and therefore more likely to cover the costs.
Get a travel insurance (valid 6-8 weeks)
This is a good option, if you are travelling for a short period of time. A travel insurance costs 15 - 25 € a year, but it is limited to 6 - 8 weeks per trip.
Get an overseas health insurance (restricted period of validity)
If you are travelling longer than 8 weeks this is an option. You need to specify the period of validity upfront (in days) and usually pay in advance for the specified time. Almost every insurance company offers an overseas health insurance. The biggest differences are the maximum period of validity and if they include USA/Canada. Of course you need to check the terms and conditions but according to my research they don't differ that much. An overseas health insurance costs 1 - 2 € per not including USA/Canada and around 3 € including it.
Our solution? We wanted to do a bit of a trick and get a travel insurance covering the first 8 weeks before paying for an overseas health insurance. But as you have a legal obligation for a health insurance in Germany, we cannot set our local health insurance on hold with just a travel insurance (those usually don't cover any costs in your home country). We will therefore set our local insurance on hold (saves around 500€/month) and get an overseas health insurance for around 50€/month.