Ion temperatures serve as an important diagnostic for inertial confinement fusion (ICF) implosions. In direct-drive ICF experiments on OMEGA, neutron-time-of-flight (nTOF) data are used to infer the ion temperature of the fusing plasma produced in the implosion experiment. The analysis of the nTOF data requires an assumption about the shape of the underlying source signal. Since the source nTOF signal is a near-replica of the neutron energy spectrum, an ideal Gaussian shape, corresponding to the neutron energy spectrum of a uniform temperature plasma, is routinely employed. However, spatial and temporal variations of the ion temperature in the plasma give rise to higher-order moments, which were first described by Munro [Nucl. Fusion 56, 036001 (2016)]. In this work, we show a simpler alternative analysis to derive moments of the neutron energy spectrum for a plasma with variations in ion temperature. We also present a revised analysis of measured nTOF signals that uses a model with an additional degree of freedom to take into account the effect of ion temperature variations on the shape of the spectrum. Compared to presently used nTOF analysis, the revised analysis yields on average ≈2× more accurate fits to the data and up to 15% higher ion temperatures for cryogenic experiments. Furthermore, we quantify the ion temperature inflation caused by radially symmetric fluid flows, which are present even in a symmetric implosion, and which serve as a lower bound on the ion temperature inflation in real implosions.