We present the results of an improved Monte Carlo Glauber (MCG) model of relevance for collisions involving nuclei at center-of-mass energies of BNL RHIC ($sqrt{s_{rm NN}}=0.2$ TeV), CERN LHC ($sqrt{s_{rm NN}}=2.76$-$8.8$ TeV), and proposed future hadron colliders ($sqrt{s_{rm NN}}approx 10$-$63$ TeV). The inelastic pp cross sections as a function of $sqrt{s_{rm NN}}$ are obtained from a precise data-driven parametrization that exploits the many available measurements at LHC collision energies. We describe the nuclear transverse profile with two separated 2-parameter Fermi distributions for protons and neutrons to account for their different densities close to the nuclear periphery. Furthermore, we model the nucleon degrees of freedom inside the nucleus through a lattice with a minimum nodal separation, combined with a recentering and reweighting procedure, that overcomes some limitations of previous MCG approaches. The nuclear overlap function, number of participant nucleons and binary nucleon-nucleon collisions, participant eccentricity and triangularity, overlap area and average path length are presented in intervals of percentile centrality for lead-lead (PbPb) and proton-lead (pPb) collisions at all collision energies. We demonstrate for collisions at $sqrt{s_{rm NN}}=5.02$ TeV that the central values of the Glauber quantities change by up to 7%, in a few bins of reaction centrality, due to the improvements implemented, though typically remain within the previously assigned systematic uncertainties, while their associated uncertainties are generally smaller (mostly below 5%) at all centralities than for earlier calculations. Tables for all quantities versus centrality at present and foreseen collision energies involving Pb nuclei, as well as for collisions of XeXe at $sqrt{s_{rm NN}}=5.44$, and AuAu and CuCu at $sqrt{s_{rm NN}}=0.2$ TeV, are provided.
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