Numerical experiments show an accuracy degeneracy phenomena in essentially nonoscillatory (ENO) finite difference schemes for hyperbolic conservation laws [cf. \textit{A. M. Rogerson} and \textit{E. Meiburg}, ibid. 5, No.2, 151-167 (1990; Zbl 0732.65086)]. The idea of using adaptive stencils forms the very basis of such schemes, but it seems to be attractive to develop modified schemes that employ fixed stencils in smooth regions and adaptive stencils in regions of steep gradients. An in this way modified scheme recovers the correct order of accuracy for all test problems with smooth initial conditions and gives results comparable to the original schemes for discontinuous problems.