This paper presents a diagonal-secant modification of the successive element correction method, a finite-difference based method, for sparse unconstrained optimization. This new method uses the gradient values more efficiently in forming the approximate Hessian than the successive element correction method. It is shown that the new method has at least the same local convergence rates as the successive element correction method for general problems and that it has better q-convergence and f-convergence rates than the successive element correction method for problems with band structures. The numerical results show that the new method may be competitive with most of the existing methods for some problems.