We consider the quantum Ising chain with uniformly distributed random antiferromagnetic couplings (1 ≤ Ji ≤ 2) and uniformly distributed random transverse fields (Γ₀ ≤ Γ_i ≤ 2Γ₀) in the presence of a homogeneous longitudinal field, h. Using different numerical techniques (density-matrix renormalization group, combinatorial optimization, and strong disorder renormalization group methods) we explore the phase diagram, which consists of an ordered and a disordered phase. At one end of the transition line (h = 0, Γ₀ = 1) there is an infinite disorder quantum fixed point, while at the other end (h = 2, Γ₀ = 0) there is a classical random first-order transition point. Close to this fixed point, for h > 2 and Γ₀ > 0 there is a reentrant ordered phase, which is the result of quantum fluctuations by means of an order through disorder phenomenon.