Using the corner-transfer matrix renormalization group to contract the tensor network that describes its partition function, we investigate the nature of the phase transitions of the hard-square model, one of the exactly solved models of statistical physics for which Baxter has found an integrable manifold. The motivation is twofold: assess the power of tensor networks for such models, and probe the 2D classical analog of a 1D quantum model of hard-core bosons that has recently attracted significant attention in the context of experiments on chains of Rydberg atoms. Accordingly, we concentrate on two planes in the 3D parameter space spanned by the activity and the coupling constants in the two diagonal directions. We first investigate the only case studied so far with Monte Carlo simulations, the case of opposite coupling constants. We confirm that, away and not too far from the integrable 3-state Potts point, the transition out of the period-3 phase appears to be unique in the Huse-Fisher chiral universality class, albeit with significantly different exponents as compared to Monte Carlo. We also identify two additional phase transitions not reported so far for that model, a Lifshitz disorder line, and an Ising transition for large enough activity. To make contact with 1D quantum models of Rydberg atoms, we then turn to a plane where the ferromagnetic coupling is kept fixed, and we show that the resulting phase diagram is very similar, the only difference being that the Ising transition becomes first-order through a tricritical Ising point, in agreement with Baxter's prediction that this plane should contain a tricritical Ising point, and in remarkable, almost quantitative agreement with the phase diagram of the 1D quantum version of the model.