We extend the strong macroscopic stability introduced in Bramson and Mountford [Ann. Probab. 30 (2002) 1082-1130] for one-dimensional asymmetric exclusion processes with finite range to a large class of one-dimensional conservative attractive models (including misanthrope process) for which we relax the requirement of finite range kernels. A key motivation is the extension of constructive hydrodynamics result of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1-30, Ann. Probab. 34 (2006) 1339-1369, Electron. J. Probab. (to appear)] to nonfinite range kernels.