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We present black hole solutions in $2+1-$dimensional Einstein's theory ofgravity coupled with Born-Infeld nonlinear electrodynamic and a masslessself-interacting scalar field. The model has five free parameters: mass $M$,cosmological constant $\ell $, electric $q$ and scalar $r_{0}$ charges andBorn-Infeld parameter $\beta $. To attain exact solution for such a highlynon-linear system we adjust, i.e. finely tune, the parameters of the theorywith the integration constants. In the limit $\beta \rightarrow 0$ we recoverthe results of Einstein-Maxwell-Scalar theory, obtained before. The selfinteracting potential admits finite minima apt for the vacuum contribution.Hawking temperature of the model is investigated versus properly tunedparameters. By employing this tuned-solution as basis, we obtain also a dynamicsolution which in the proper limit admits the known solution in Einsteingravity coupled with self-interacting scalar field. Finally we establish theequations of a general scalar-tensor field coupled to nonlinear electrodynamicsfield\ in $2+1-$dimensions without searching for exact solutions.

Einstein-Born-Infeld black holes with a scalar hair in three-dimensions