A new method is presented for enclosing the reachable sets of nonlinear ordinary differential equations subject to a range of inputs. Reachable set enclosures are used for uncertainty propagation, robust control, and global optimization of dynamic systems arising in a variety of applications. However, existing methods often provide an unworkable compromise between cost and accuracy. For example, fast interval methods often produce divergent bounds, while methods based on more complex sets scale poorly with problem size. To overcome this, a novel method is introduced for reducing the conservatism of fast interval methods through the select addition of redundant model equations that can be exploited in the bounding procedure. Several case studies demonstrate that such redundancy can dramatically reduce conservatism. The additional cost is modest in most cases, but does become significant when many redundant equations are used. (C) 2017 Elsevier Ltd. All rights reserved.