The answer is no. If $L$, $L^\prime$ are 3-dimensional lens space and $S^1\times L$ is diffeomorphic to $S^1\times L^\prime$, then the covering space of $S^1\times L$ corresponding to the torsion subgroup defines an h-cobordism between $L$ and $L^\prime$. It is an application of Atiyah-Singer fixed point theorem (with contributions by Bott and Milnor), that h-cobordant lens spaces are diffeomorphic. One reference is p.479 in "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications" by Atiyah and Bott.
Various related results and generalizations are discussed in "Toral and exponential stabilization for homotopy spherical spaceforms" by Kwasik and Schultz. Both papers can be easily found online, I think.