The Barycentric Lagrange interpolation with non-uniformly spaced Chebyshev interpolation nodes has been modified and re-configured in a new formula to solve Volterra integral equations of the second kind. To avoid the application of the collocation methods necessary to transform the solution into an equivalent linear system, the unknown function is interpolated using the given modified Barycentric Lagrange polynomials and is substituted twice into the integral equation, while the given data function is expanded into Maclaurin polynomial of the same degree as the interpolant unknown function. Thus, the presented technique yields substantially higher accuracy and faster convergence of the solutions. The obtained results of the five illustrated examples show that the numerical solutions converge to the exact solutions, particularly for unknown analytic functions, and strongly converge to the exact ones for all values of  including the endpoints of the integration domain regardless of whether the kernel and data functions are algebraic or not.