<p>A new scheme of approximate solution of the problem of axial compression of an elastic cylinder with one movable and the other fixed end with a free lateral surface is presented and refines the known solution obtained using separation of variables when averaging conditions over stresses on the lateral surface of the cylinder. The refinement is made by successive removal of discrepancies: first, in the stress distributions on the lateral surface of the cylinder, then in the radial displacements along the ends and further in the axial displacement of the movable end. Comparison with the results of numerical solution of the problem by the finite element method for different values of the Poisson ratio and different combinations of overall dimensions of the cylinder has shown the effectiveness of the proposed approach.</p>