  We construct new covers of the Suzuki and Ree curves which are maximal with
respect to the Hasse-Weil bound over suitable finite fields. These covers are
analogues of the Giulietti-Korchm\'aros curve, which covers the Hermitian curve
and is maximal over a base field extension. We show that the maximality of
these curves implies that of certain ray class field extensions of each of the
Deligne-Lusztig curves. Moreover, we show that the Giulietti-Korchm\'aros curve
is equal to the above-mentioned ray class field extension of the Hermitian
curve.
