We have recently proposed a new technique of plasma tailoring by laser-driven hydrodynamic shockwaves generated on both sides of a gas jet [Marquès et al., Phys. Plasmas 28, 023103 (2021)]. In a continuation of this numerical work, we study experimentally the influence of the tailoring on proton acceleration driven by a high-intensity picosecond laser in three cases: without tailoring, by tailoring only the entrance side of the picosecond laser, and by tailoring both sides of the gas jet. Without tailoring, the acceleration is transverse to the laser axis, with a low-energy exponential spectrum, produced by Coulomb explosion. When the front side of the gas jet is tailored, a forward acceleration appears, which is significantly enhanced when both the front and back sides of the plasma are tailored. This forward acceleration produces higher-energy protons, with a peaked spectrum, and is in good agreement with the mechanism of collisionless shock acceleration (CSA). The spatiotemporal evolution of the plasma profile is characterized by optical shadowgraphy of a probe beam. The refraction and absorption of this beam are simulated by post-processing 3D hydrodynamic simulations of the plasma tailoring. Comparison with the experimental results allows estimation of the thickness and near-critical density of the plasma slab produced by tailoring both sides of the gas jet. These parameters are in good agreement with those required for CSA.