Here, the Cos and Sin lines are colored the same as the fill, and they lie below the other line, if that is what you mean by “border”.
Show[Plot[{Cos[x], Sin[x]}, {x, 0, 2 Pi}, Filling -> {1 -> {2}}, FillingStyle -> Directive[LightGray], PlotStyle -> LightGray], Plot[Sech[x], {x, 0, 2 Pi}]]

Adding
Thanks for agreeing. Here's a more complete answer, after your editing:
Show[Plot[{Sech[x], If[Sech[x] > 0.8, 0.8], If[Sech[x] > 0.5, 0.5], If[Sech[x] > 0.3, 0.3], If[0.5 > Sech[x] > 0.3, Sech[x]]}, {x, 0, 2 Pi}, Filling -> {1 -> {2}, 3 -> {4}, 4 -> {5}}, FillingStyle -> LightGray, PlotStyle -> LightGray], Plot[Sech[x], {x, 0, 2 Pi}, PlotStyle -> Gray], Plot[If[0.3 > Sech[x], Sech[x]], {x, 0, 2 Pi}, PlotStyle -> Directive[{Thick, Black}]], Plot[If[0.8 > Sech[x] > 0.5, Sech[x]], {x, 0, 2 Pi}, PlotStyle -> Directive[{Thick, Black}]], Map[Graphics[Style[Line[{ {ArcSech[
