We consider the sparse linear regression model 𝐲 = X β +𝐰, where X ∈ℝ^n × d is the design, β∈ℝ^d is a k-sparse secret, and 𝐰∼ N(0, I_n) is the noise. Given input X and 𝐲, the goal is to estimate β. In this setting, the Lasso estimate achieves prediction error O(k log d / γ n), where γ is the restricted eigenvalue (RE) constant of X with respect to support(β). In this paper, we introduce a new semirandom family of designs – which we call partially-rotated designs – for which the RE constant with respect to the secret is bounded away from zero even when a subset of the design columns are arbitrarily correlated among themselves. As an example of such a design, suppose we start with some arbitrary X, and then apply a random rotation to the columns of X indexed by support(β). Let λ_min be the smallest eigenvalue of 1/n X_support(β)^⊤ X_support(β), where X_support(β) is the restriction of X to the columns indexed by support(β). In this setting, our results imply that Lasso achieves prediction error O(k log d / λ_min n) with high probability. This prediction error bound is independent of the arbitrary columns of X not indexed by support(β), and is as good as if all of these columns were perfectly well-conditioned. Technically, our proof reduces to showing that matrices with a certain deterministic property – which we call restricted normalized orthogonality (RNO) – lead to RE constants that are independent of a subset of the matrix columns. This property is similar but incomparable with the restricted orthogonality condition of [CT05].