Here is a solution with NDSolve components:

state = First@NDSolve`ProcessEquations[{x''[s]+x[s]==0,x[0]==0,x'[0]==1},x,s];
data=Rest@Flatten@Reap@Do[
    NDSolve`Iterate[state,0.1];
    point=NDSolve`SolutionDataComponent[state@"SolutionData"["Forward"], "DependentVariables"];
    Sow[{t,point}];
    state=First@NDSolve`Reinitialize[state,Thread[{x[0],x'[0]}==point]];,
    {t,0.1,1.,0.1}
]
Show[Plot[Sin[s],{s,0,1},Frame->True,Axes->False],ListPlot[Partition[data,3][[;;,1;;2]],PlotStyle->{Red,PointSize[Large]}]]

Hi,

Try to use "EventLocator" method for NDSolve (or WhenEvent[] option). Here is an example with "EventLocator" (for some reason values at 0. are not catched):

data = Flatten[
    Reap[
       NDSolve[
         {
         x''[s]+x[s]==0,
         x[0]==0,
         x'[0]==1
         },
         {},
         {s,0.,1.},
         Method->{
          "EventLocator",
          "Event"-> Thread[s-Table[x,{x,0.,1.,0.2}]],
          "EventAction" :> Sow[{s,x[s],x'[s]}],
          "Direction" -> 1
         }
       ]
    ]
]
Show[
    Plot[Sin[s],{s,0,1},Frame->True,Axes->False],
    ListPlot[Partition[data,3][[;;,1;;2]],PlotStyle->{Red,PointSize[Large]}]
]

Not sure if it is possible with NDSolve components, you can try to set EventLocator method for NDSolve`ProcessEquations with SetOptions[]