hilbertTransformSmoothFinite[f_, x_?NumericQ, supp_, s_: 1] :=
 Module[{esupp, ef},
  esupp = {supp[[1]] - Abs[f[supp[[1]]]]/s,
    supp[[2]] + Abs[f[supp[[2]]]]/s};
  ef = Function[X, Piecewise[{{f[X], supp[[1]] < X <= supp[[2]]},
      {f[supp[[1]]] + Sign[f[supp[[1]]]] s (X - supp[[1]]),
       esupp[[1]] < X <= supp[[1]]},
      {f[supp[[2]]] - Sign[f[supp[[2]]]] s (X - supp[[2]]),
       supp[[2]] < X <= esupp[[2]]}, {0, True}}]];
  NIntegrate[
    1/\[Pi] (ef[\[Xi]] - ef[x])/(\[Xi] - x), {\[Xi], esupp[[1]],
     esupp[[2]]}] +
   1/\[Pi] ef[x] Log[(esupp[[2]] - x)/(x - esupp[[1]])]]
hh = h /. NDSolve[{h'[x] == x, h[0] == 0}, h, {x, 0, 1}][[1]];
p = Function[x, hilbertTransformSmoothFinite[hh, x, {0, 1}]];
Plot[p[x], {x, 0, 1}]
Plot[p'[x], {x, 0, 1}]