We discuss rigid compact complex manifolds of Kodaira dimension 1, arising as product–quotient varieties. First, we show that there is no free rigid action on the product of \((n-1)\) elliptic curves and a curve of genus at least 2. Then, we describe the occurring groups, study the quotients, and prove that there is always a suitable resolution of singularities preserving rigidity. Finally, we give a complete classification of the cases arising for minimal group orders.