Trademark

Adding "The" does not change the mark enough to allow use if it infringes. Nevertheless, if you did not register the mark in that state of federally, then your protection would only derive from use of the mark in that state. If you do business in that other state and established the mark as being identified with your business, then you may have some rights.

You should register federally just as an added protection.

Your first question should be whether or not you have a federal trademark registration for your brand name. If the answer is no, then you will have a significantly harder time enforcing your brand name rights outside of your state. Secondly, you must ask if the the other business is in the same products/services space as yourself. For while, generally speaking, distinguishing two marks with "the" will not be a defense against infringement, you must assess whether or not the use the mark by the other company is actually infringing. For example, although "Kodak" is associated strongly with optics and photography products, a small dog walking business called "Kodak" may possibly not be infringing This is because there is no "likelihood of confusion," which means that the public at large would not likely believe that Kodak got into the pet care business.

So you should likely consult a local attorney to analyze these and other issues to see what your options are.

I hope this helps.

Disclaimer: This answer is for informational purposes only and does not constitute general or specific legal advice, nor create an attorney client relationship.