sol = NDSolveValue[{y'[t] == y[t], y[0] == 1}, y, {t, 0, 1},
InterpolationOrder -> All, Method -> "ExplicitRungeKutta"];
{{grid}, coeff, {{idcs, types}}} = (List @@ sol)[[3 ;;]];
y0 = Piecewise[#, Indeterminate] &@Quiet[
Map[
{chebeval[coeff[[Last@#1]]][Rescale[t, grid[[#1]], {-1, 1}]],
grid[[First@#1]] <= t <= grid[[Last@#1]]} &,
Rest@idcs],
CompiledFunction::cfsa]