If two K3 surfaces X and Y over C admit a rational map of finite degree X→Y, Inose proved that their Picard numbers ρ(X) and ρ(Y) are equal. Suppose X admits an elliptic fibration π:X→P1. By a base change b:P1→P1, we obtain another elliptic surface π×b:X′:=X×P1P1→P1. If X′ is once again a K3 surface, we know ρ(X′)=ρ(X). However, it is difficult in general to find generators of the N\'eron-Severi goup of X′. Starting from various K3 surfaces X having a Shioda-Inose structure, we construct X′→P1 whose Mordell-Weil rank is large, and explore methods of finding generators of the Mordell-Weil group.