This study presents a refined calculation of the atmospheric neutrino flux from GeV to PeV energies based on two data-driven models for the flux of cosmic ray nucleons (GSF6) and hadronic interactions (DDM, previously detailed in Fedynitch & Huber, PRD106 083018, 2022). The uncertainties are treated as adjustable parameters and are fitted using a combination of muon spectrometer data at the surface, including constraints from fixed-target experiments. The resulting calculated neutrino fluxes have uncertainties of less than 10% up to 1 TeV and show significant differences compared to previous calculations. We discuss which experiments have the most significant impact on the fit and propose how this method can incorporate additional constraints, such as measurements of deep underground muon intensities or seasonal variations, in the future. Our model is publicly available via the daemonflux code, which provides access to all model parameters and the covariance matrix obtained from the fit.