Generally speaking, yes, having a trademark registered in more classes means broader legal protection. It gives you a stronger case when challenging new applications in your classes as well as dealing with infringement attempts from brands offering goods/services similar to what is listed in your application.
However, there are two important prerequisites for acquiring such protection:
- The classes have to describe goods/services actually offered by your brand. You can theoretically register in classes where you wish to expand to with your businesses, but if you fail to do so or stop providing said goods/services at some point, it can be challenged by a third party. In the US specifically, you have to provide proof of entering the US market before your registration can be finalized. After that, when submitting the Declaration of Use (between the 5th and 6th year) or a renewal request (between the 9th and 10th year), you may be asked by the USPTO to limit the selection of your classes if you are not commercially active in them for a certain period of time.
- You have to pass the opposition window, i.e. either not receive or resolve any opposition raised against your trademark application in all your listed classes. And there is where it can get tricky as more classes mean more trademark owners can come forward and oppose your mark based on the perceived similarity, where class similarity plays a role.
Furthermore, keep in mind that each additional class listed on the application will cost extra.