  We consider higher-order derivative gauge field corrections that arise in the
fundamental context of dimensional reduction of String Theory and
Lovelock-inspired gravities and obtain an exact and asymptotically flat
black-hole solution, in the presence of non-trivial dilaton configurations.
Specifically, by considering the gravitational theory of Euler-Heisenberg
non-linear electrodynamics coupled to a dilaton field with specific coupling
functions, we perform an extensive analysis of the characteristics of the black
hole, including its geodesics for massive particles, the energy conditions,
thermodynamical and stability analysis. The inclusion of a dilaton scalar
potential in the action can also give rise to asymptotically (A)dS spacetimes
and an effective cosmological constant. Moreover, we find that the black hole
can be thermodynamically favored when compared to the
Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black hole for those
parameters of the model that lead to a larger black-hole horizon for the same
mass. Finally, it is observed that the energy conditions of the obtained black
hole are indeed satisfied, further validating the robustness of the solution
within the theoretical framework, but also implying that this self-gravitating
dilaton-non-linear-electrodynamics system constitutes another explicit example
of bypassing modern versions of the no-hair theorem without any violation of
the energy conditions.
