For a particular family of pairs of explicit Runge–Kutta methods of orders $p - 1$ and p, sets of efficient, continuously differentiable interpolants of several orders up to p are characters zed algorithmically in terms of several arbitrary parameters. The approach can be applied in an obvious way to yield interpolants for other types of explicit Runge–Kutta methods, as well as families of other types of explicit and implicit methods. Derivative evaluations required for each pair of methods are reused, and additional derivative evaluations are selected in an attempt to minimize the total number of stages required. The analysis provides a lower bound on the number of stages required, and indicates, for example, why twelve stages are required to provide interpolants for eight-stage pairs of orders 5 and 6. In contrast to the eighteen stages used to obtain a known interpolant of order 7 for a pair of methods of orders 6 and 7, only sixteen stages are required by the proposed derivation. Details for interpolants of orders $p = 8$, and $p = 9$ are also given. However, it has not been established that the lower bound is sharp, so further improvement may be possible.