A new exponentially fitted finite difference method is introduced for the numerical treatment of a second-order singularly perturbed differential equation (SPDDE) involving small delays in the first derivative and undifferentiated terms. The solution of such equations exhibits left-layer or right-layer behavior in the underlying domain. Taylor’s series expansion procedure is used for constructing an equivalent valid version of the original problem and deriving a new three-term finite difference scheme, respectively. The non-uniformity in the solution behavior is resolved by introducing an appropriate exponential fitting parameter in the derived new scheme. The resulting system of equations is solved by the well-known ’discrete invariant algorithm.’ Convergence analysis of the fitted method is discussed, and the theory is illustrated by performing numerical experiments on test example problems. Tabulated computational results show the applicability and accuracy of the method. Theory and computation show that the method is able to approximate the solution very well with a second-order convergence rate. The graphical representation of the solution graph for the tested problems illustrates the impact of varied delay shifts on the layer behavior of the solution.