This paper considers the enhancement of a given Runge-Kutta method by the addition of an interpolation formula. The interpolation formula proposed makes use of the same derivative values as are used in the main method. It also possibly uses the derivative found at the end of the step and the derivative at an additional, appropriately chosen off-step point. The aim is to maintain an accuracy for interpolated results that is as close as possible to that of the principal method on which it is based. Such extended Runge-Kutt- a methods as these can be used to provide dense output and to enable discontinuities to be located efficiently. For three representative methods, each of which is already provided with a built-in error estimator, formulas are found for providing this interpolation and each extended method is tested with a collection of standard problems. At best, the results achieved are all that could be hoped for, with off-step interpolated results of comparable accuracy to that of the main method. However, there is a wide spread of observed behaviors; at worst, the errors can be much larger than those at the step values.