This paper describes the design and construction of an extended-range multisphere neutron spectrometer (ERMNS), also called a Bonner sphere spectrometer, created to measure the spectrum of cosmic-ray-induced neutrons on container ships. Such measurements require a highly sensitive neutron spectrometer and can benefit from improved energy resolution. To obtain high sensitivity, spherical 3 He gas proportional counters with a diameter of 152 mm (6 in.) were developed and used. The goal for designing the moderator assemblies of the new ERMNS was to optimize its energy resolution where it is typically worst for such spectrometers - at high and medium-low energies. Absorber shells containing boron carbide were used to improve resolution at medium-low energies. The response functions of various sizes and designs of plain and modified spheres were calculated using the radiation transport code MCNPX 2.6b. The resolution of combinations of 16 spheres of candidate designs was then determined using Reginatto's RESPOW code, and the sphere designs that gave the smallest standard deviations of the RESPOW averaging kernels were selected. Compared to an ERMNS used earlier for similar measurements, resolution improved by 7 to 17% from 10 eV to 0.1 MeV and 24 to 39% from 10 MeV to 10 GeV. Bayesian parameter estimation was also used to characterize the uncertainties in spectra measured with the old and new spectrometers. The uncertainties in the fluence rates and peak energies of a typical terrestrial cosmic-ray neutron spectrum improved by 15 to 85% and 56 to 400%, respectively. These results demonstrate the utility of RESPOW and Bayesian parameter estimation for designing ERMNSs.