Some insurance is more insuring than other insurance. First you have to consider what you are getting insured for (e.g. travel, liability, health). Travel insurance insures you for travel costs. According to my understanding whether you are teaching or not has no impact on your travel insurance. Once you are in the US, travel insurance should only cover expenses related to travelling back home (e.g. if you got injured or sick and out for the season and needed to change your flight home and that cost extra $$). Liability insurance can be useful if one of your students gets hurt while you are teaching. Most US instructors rely on their employers for this insurance. But there is a small chance that an instructor could be held personally liable, so a very small number of instructors get insured because they have personal assets that are worth going after (I don't know anyone who does, but I've seen ads selling this). Most instructors in the US who buy personal liability insurance buy it because they teach in "informal" situations (e.g. camps/clinics). Health insurance comes in two flavors: workers' compensation and regular insurance. Workers' comp only covers accidents that occur while working, but it covers health costs and wage replacement costs. Workers' comp is paid for by the employer. Workers' comp coverage typically requires you to go to specific health care providers. Many instructors don't care about the wage replacement aspect of workers' comp and don't want the hassle of using a health care provider who is not their regular doctor. In those cases, they just use their regular health insurance. In the past, most ski instructors got their regular insurance from their "real" "other" job. If one did not have another it was easy to get covered for everything but pre-existing conditions, but expensive to get insurance "on your own". Now, under the affordable care act (aka ObamaCare), it is easy to get covered for everything, but the paperwork/process is typically a pain. At least the costs for ObamaCare are lower and in many case they are subsidized. Some temporary immigrants may want to take advantage of ObamaCare while they are in the US (check here for eligibility). Some may want to just use their own insurance from their home country. Blue Cross/Blue Shield is widely accepted in the US and is globally available.
Some places "take" only certain kinds of health insurance. That means they will only bill you for the portions of their bill that are not covered by insurance (e.g. the copay). If you don't have insurance from companies they "like", then you have to pay the full bill and then get reimbursed by your insurance company. In that case if there is any argument about what expenses are "reasonable", you get to be in the middle of the argument. Some kinds of health insurance have "in network" and "out of network" providers. If the provider you choose in is not "in network", then your expenses are typically higher and the paperwork may be greater. Some insurance companies provide coverage for when you travel out of your home country. You may be covered in the US under health insurance acquired in Australia. Even if you are, it may still be a benefit to acquire additional coverage here.
Deciding what option is right for you is part of the "joy" of the process. Some people might even call it a "blessing". Others might call it things we can't politely say. Hope this helps.