By using extensive tensor network calculations, we map out the phase diagram of the frustrated J_{1}-J_{2} Ising model on the square lattice. In particular, we focus on the cases with controversy in the phase diagram, especially the stripe transition in the regime g=|J_{2}/J_{1}|>1/2 (J_{2}>0,J_{1}<0). While recent studies claimed that the phase transition is of first order when 1/2<g<g^{*} (with the smallest g^{*} being 0.67), our simulations suggest that if there is such a first-order region, it is smaller than those found in earlier studies by other methods. Combining with the analysis of critical properties, we provide evidence that the classical J_{1}-J_{2} model evolves continuously from two decoupled Ising models (g→∞ with central charge c=1) to a point belonging to the tricritical Ising universality class (with c=0.7) as g decreases to g^{*}≃0.54.