In this study, we present a two-step reconstruction approach for mapping glacier thickness that solves mass conservation over single or several connected drainage basins. The approach is applied to a variety of test geometries with abundant thickness measurements including marine- and land-terminating glaciers as well as an ice cap on Svalbard. In the first step, a geometrically controlled, non-local flux solution is converted into thickness values relying on the shallow ice approximation (SIA). In a second step, the thickness field is updated along fast-flowing glacier trunks on the basis of velocity observations. Both steps account for available thickness measurements. Each thickness field is presented together with an error-estimate map based on a formal propagation of input uncertainties. For Vestfonna ice cap, a previous ice volume estimate based on the same measurement record as used here has to be corrected upward by 22 %. We also find that a 13% area-fraction of the ice cap is in fact grounded below sea level. The former 5%-estimate from a direct measurement interpolation exceeds the aggregate error range of 6–23%.