Finding the x coordinate of a point in the solution of NDSolve from its y coordinate

In general, there may be multiple values of x for a given value of y. For example,

Clear["Global`*"]

ysol = NDSolveValue[
  {y'[x] == y[x] Cos[x + y[x]], y[0] == 1}, y, {x, 0, 30}];

Let the value of y of interest be 0.2

Plot[{ysol[x], 0.2}, {x, 0, 30},
 PlotRange -> {0, 0.45},
 AxesLabel -> {"x", "y"}]

Since the inverse is multi-valued, an estimate for x must be provided to indicate which solution should be found.

xval[y_?NumericQ, xEst_?NumericQ] := 
  FindRoot[ysol[x] == y, {x, xEst}]

To find all solutions,

xsol = x /. (xval[0.2, #] & /@ {3, 7, 9, 13, 15})

(* {3.03258, 6.20767, 8.80558, 13.1243, 14.6631} *)

Verifying that these values of x give the target value

And @@ (ysol[#] == 0.2 & /@ xsol)

(* True *)