In　the　present　paper,　the　author　investigates　the　initial　problem　for　genuinely　nonlinear　quasilinear　hyperbolic　systems　of　conservation　laws　under　small　BV　perturbations　of　the　Riemann　initial　data,　where　the　perturbations　are　in　BV　but　they　are　assumed　to　be　C-1-smooth,　with　bounded　and　possibly　large　C-1-norms.　Combining　the　techniques　employed　by　Li　and　Kong　with　the　modified　Glimm＇s　functional,　the　author　obtains　the　almost　global　existence　and　lifespan　of　classical　discontinuous　solutions　to　the　above　problem.　This　result　is　also　applied　to　the　system　of　one-dimensional　isentropic　flow.　(C)　2013　Elsevier　Inc.　All　rights　reserved.