Graphs whose spectrum belongs to the interval $[-2,2]$ are called Smith graphs. Vertices of the cospectrality graph $C(H)$ of a Smith graph $H$ are all graphs cospectral with $H$ with two vertices adjacent if there exists a certain transformation transforming one to another. We study how the cospectrality graph of the union of two Smith graphs can be composed starting from cospectrality graphs of starting graphs.