Can a person that has been refused entry to a Schengen country, re-enter to other Schengen country, if no visa is required?

You can most likely enter. A ban is distinct from a mere entry refusal and would only be imposed if you did something serious like using a forged passport, lying to get a visa or working illegally (although there are differences between countries in exactly how serious an infringement needs to be to warrant a ban). Unless you were told you have also been banned from entry (and for how long), there is no reason to believe you have been banned. And unless there is such a ban, a prior refusal does not formally prevent border guards from granting entry if they are satisfied that you now fulfil the requirements.

At the moment, the previous incident wouldn't even be recorded in any Schengen-wide database but you might however get questioned, e.g. based on the stamps on your passport. If that happens, additional documentation (bookings, proof of financial means, etc.) could come in handy. You don't need a visa anymore but, like all visitors from outside the EU, you have to fulfil most of the same requirements (except travel health insurance). Border guards rarely check them systematically for people who don't need a visa but they are legally entitled to and may very well refuse entry based on that.