We consider the family of one-frequency quasi-periodic Schrödinger coycles Gω,E​, parametrized by the energy E. For potential functions v(x)=λv0​(x), where v0​∈C2(T,R) is a Morse function with finitely many critical points and λ>0 is large, we show that, for any value of E∈R and for any phase x∗∈T such that ∣v0​(x∗)−E/λ∣ is not too small, there exists a set of frequencies Ω=Ω(E,x∗) of positive measure such that the following hold: (1) for every ω∈Ω, the upper Lyapunov exponent of the cocycle Gω,E​ is ≳logλ and x∗ is (essentially) a typical point in Oseledets’ theorem; (2) either Gω,E​ is uniformly hyperbolic, or there exists a phase x0​∈T such that E is an eigenvalue of the corresponding discrete Schrödinger operator Hx0​,ω​.