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Contemporary ultraintense, short-pulse laser systems provide extremely compact setups for the production of high-flux neutron beams, such as those required for nondestructive probing of dense matter, research on neutron-induced damage in fusion devices or laboratory astrophysics studies. Here, by coupling particle-in-cell and Monte Carlo numerical simulations, we examine possible strategies to optimise neutron sources from ion-induced nuclear reactions using 1-PW, 20-fs-class laser systems. To improve the ion acceleration, the laser-irradiated targets are chosen to be ultrathin solid foils, either standing alone or preceded by a plasma layer of near-critical density to enhance the laser focusing. We compare the performance of these single- and double-layer targets, and determine their optimum parameters in terms of energy and angular spectra of the accelerated ions. These are then sent into a converter to generate neutrons via nuclear reactions on beryllium and lead nuclei. Overall, we identify configurations that result in neutron yields as high as $\sim 10^{10}\,\rm n\,sr^{-1}$ in $\sim 1$-cm-thick converters or instantaneous neutron fluxes above $10^{23}\,\rm n\,cm^{-2}\,s^{-1}$ at the backside of $\lesssim 100$-$μ$m-thick converters. Considering a realistic repetition rate of one laser shot per minute, the corresponding time-averaged neutron yields are predicted to reach values ($\gtrsim 10^7\,\rm n \,sr^{-1}\,s^{-1}$) well above the current experimental record, and this even with a mere thin foil as a primary target. A further increase in the time-averaged yield up to above $10^8\,\rm sr^{-1}\,s^{-1}$ is foreseen using double-layer targets.