We considered a situation where a Ricci scalar R and the scalar field φ are inversely coupled each other by the term 1/2ξφ^{-n}R, where ξ denotes the strength of non-minimal coupling. This term is motivated to make Einstein-frame inflationary potential flat in a small limit of scalar field value, to guarantee high number of e-folds during inflation to cure limitations that a standard Big Bang itself cannot resolve. (e.g. flatness problem, horizon problem, etc.) On the other hand, the potential V (φ) = V0φ^{−n} is one of simplest models for explaining late-time acceleration due to runaway behavior at the large field limit, which in addition possesses attractor-like solution and well motivated by particle physics models. These two facts motivated us to study a model where a scalar field φ is inversely non-minimally coupled to the gravitational sector with aforementioned runaway potential, to explain both inflationary observations and late-time cosmic acceleration. As an inflationary stage, it is confirmed that choosing most of parameter sets (V0, ξ, n) enables us to satisfy inflationary constraints given by the latest observations from Planck-BICEP/Keck. Our setup can also explain late-time acceleration (especially showing behavior like cosmological constant in a neighborhood of current Universe (w ≃ −1) and is compatible with several constraints for successful Big Bang scenario for some suitable initial conditions. We especially discussed necessary theoretical criteria for choosing adequate initial conditions by the existence of Hubble drag, which plays a role to limit the velocity of a quintessence field during its evolution.