The issue of improving the error estimate of a numerical scheme constructed to solve a certain integral equation of the second kind is considered. The equation is obtained by regularizing the Shtaerman equation of the first kind, known in elasticity theory, which describes the contact problem of two elastic cylinders with almost equal radii. The integral part of this equation contains a kernel with a weak singularity, and thus the given equation of the second kind has some features, which in general, despite its Fredholm property, complicates the direct application of known methods of approximate solution to the equation. However, based on a relatively easy to implement modification of the quadrature method, it is possible to construct a fairly effective scheme for solving the corresponding equation and obtain an error estimate indicating the order of convergence, which is actually done in [1]. The numerical scheme proposed in this article differs somewhat from the schemes usually used for Fredholm equations and, taking into account a number of properties of the equation under consideration, allows one to justify the corresponding computational process in the space H_(1/2) [-x_0,x_0] with obtaining an error estimate of the form O(n^(-1/2)). The presented work will discuss the possibility of improving this estimate.