Where as owning an aircraft has its own set of insurance challenges, you as a renter need to be insured too. At least against third party liability claims and aircraft damages.
Some aircraft rental organizations include the renter in their policies. Unless you actually read their policy there is no way you can be sure that the coverage is adequate for your type of personal flying and situation.
Therefore it is best to buy your own non-owned liability insurance and remember: it is usually applicable to private flying only.
This is a policy to aimed protect you against claims arising from bodily injury and property damage for which you are legally liable, caused by an occurrence arising from your use of a non-owned aircraft, borrowed or rented. This coverage does not apply to the non-owned aircraft you have borrowed or rented. You will have to purchase physical damage coverage to the non-owned aircraft you use.
This is important as the aircraft rental company may hold you liable for loss of income during the time that the aircraft is not able to fly.
If you purchase physical damage to your non-owned aircraft coverage, the policy will provide you with liability coverage arising from damage while using someone elses aircraft. Your non-owned coverage is not a substitute for the aircraft owner buying their own coverage to protect their interests.
There is no standard recommended amount of liability coverage you should carry. Important factors such as your personal assets, earnings, whom you carry as passengers (if any), and how much insurance you can afford and what is available dictate the amount of coverage needed.
For non-owned physical damage coverage, it depends on the value of the aircraft you typically rent and whether or not you want to fully be covered in the event of a total loss.
An owners insurance covers the owner and might not include someone flying the aircraft occasionally. If you would have an accident with damage to the aircraft you might be held liable and you would need an insurance to cover these expenses.
Non-owned insurance policies do not provide coverage to use the aircraft for non-private flying for example as: passenger or freight carrying, aerial advertising, banner or glider towing, photography or any other commercial operation (part 121, 125, 135) of any kind.
Basically, its just for private pleasure flying. No more no less.