With current observational methods it is not possible to directly measure the magnetic field in the solar corona with sufficient accuracy. Therefore, coronal magnetic field models have to rely on extrapolation methods using photospheric magnetograms as boundary conditions. In recent years, due to the increased resolution of observations and the need to resolve non-force-free lower regions of the solar atmosphere, there have been increased efforts to use magnetohydrostatic (MHS) field models instead of force-free extrapolation methods. Although numerical methods to calculate MHS solutions can deal with non-linear problems and hence provide more accurate models, analytical three-dimensional MHS equilibria can also be used as a numerically relatively "cheap" complementary method. In this paper, we present an extrapolation method based on a family of analytical MHS equilibria that allows for a transition from a non-force-free region to a force-free region. We demonstrate how asymptotic forms of the solutions can help to increase the numerical efficiency of the method. Through both artificial boundary condition testing and a first application to observational data, we validate the method's effectiveness and practical utility.