Due to that orthomorphisms have good cryptographic properties, they play an important role in cryptography for resisting cryptanalysis. A new method is presented to construct an (n+1)-bit orthomorphism from an n-bit one where n>1. First, details of the method are provided and discussed. Then a simple example is presented to show how a 2-bit orthomorphism is expanded, rearranged, and selected to form a 3-bit one. Some experimental data are analyzed to demonstrate the efficiency of the construction. The method is not only a new way to construct orthomorphisms but also a new attempt of research in orthomorphism structure for building a relationship between n-bit orthomorphisms and (n+1)-bit ones.