Given a finite, connected graph $$\mathsf {G}$$ G , the lamplighter chain on $$\mathsf {G}$$ G is the lazy random walk $$X^\diamond $$ X ⋄ on the associated lamplighter graph $$\mathsf {G}^\diamond =\mathbb {Z}_2 \wr \mathsf {G}$$ G ⋄ = Z 2 ≀ G . The mixing time of the lamplighter chain on the torus $$\mathbb {Z}_n^d$$ Z n d is known to have a cutoff at a time asymptotic to the cover time of $$\mathbb {Z}_n^d$$ Z n d if $$d=2$$ d = 2 , and to half the cover time if $$d \ge 3$$ d ≥ 3 . We show that the mixing time of the lamplighter chain on $$\mathsf {G}_n(a)=\mathbb {Z}_n^2 \times \mathbb {Z}_{a \log n}$$ G n ( a ) = Z n 2 × Z a log n has a cutoff at $$\psi (a)$$ ψ ( a ) times the cover time of $$\mathsf {G}_n(a)$$ G n ( a ) as $$n \rightarrow \infty $$ n → ∞ , where $$\psi $$ ψ is an explicit weakly decreasing map from $$(0,\infty )$$ ( 0 , ∞ ) onto [1 / 2, 1). In particular, as $$a > 0$$ a > 0 varies, the threshold continuously interpolates between the known thresholds for $$\mathbb {Z}_n^2$$ Z n 2 and $$\mathbb {Z}_n^3$$ Z n 3 . Perhaps surprisingly, we find a phase transition (non-smoothness of $$\psi $$ ψ ) at the point $$a_*=\pi r_3 (1+\sqrt{2})$$ a ∗ = π r 3 ( 1 + 2 ) , where high dimensional behavior ( $$\psi (a)=1/2$$ ψ ( a ) = 1 / 2 for all $$a \ge a_*$$ a ≥ a ∗ ) commences. Here $$r_3$$ r 3 is the effective resistance from 0 to $$\infty $$ ∞ in $$\mathbb {Z}^3$$ Z 3 .