If you exactly copied someone else's t-shirt design and added a belt to it, that would be infinrging on that design by making a "derivative work" out of it, and only the copyright owner has the right to do that. Please see the attached booklet on derivative works.
The Copyright Office answers the question "How much do I have to change in order to claim copyright in someone else's work?" with "Only the owner of copyright in a work has the right to prepare, or to authorize someone else to create, a new version of that work. Accordingly, you cannot claim copyright to another's work, no matter how much you change it, unless you have the owner's consent."
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Assuming the existing product is patented, it would be an infringement of that patent to make, use, or sell a product that is created simply by adding something to the existing product.
For example: Existing Product consists of elements A, B, and C. New Product also consists of A, B, and C but also has D. New Product is a "new" product (and may even be patentable) but it also infringes the patent that protects Existing Product because New Product has all the elements that make up Existing Product. If the creator of New Product wants to make, use, or sell New Product it must first buy a license from the owner of the patent that protects the Existing Product.
If the Existing Product is not patent protected it may be protected under trade dress law. In short, if the nonfunctional features of a product's design informs consumers where the product came from, then that design is protectable trade dress (think of a glass Coke bottle -- w/o seeing the brand name "Coke" the shape of the bottle tells us that the cola inside the bottle is Coke). A very fact-based analysis would be needed to determine if your added "something" changes the appearance of the product sufficiently to alter the consumer's perception of the product.