  We investigate novel exact solutions to an Einstein-Maxwell theory
non-minimally coupled to a self-interacting dilaton-like scalar. Extending the
results of arXiv: 2012.13412 and 2110.04445, we report three families of exact
configurations over a non-relativistic Schr\"odinger background with both,
arbitrary dynamical critical exponent $z$ and hyperscaling violating parameter
$\theta$. Concretely, we provide field configurations with hyperscaling
violation which are asymptotically Schr\"odinger spaces. Our solutions
correspond to three kinds: a zero-temperature background, a naked singularity
and, more interestingly, a family of black holes. To the later, we construct
the corresponding Carter-Penrose diagram with a view to understand their causal
structure given the non-standard background. We show that a non-trivial
hyperscaling violation parameter $\theta$ is necessary in order to support a
real non-constant dilaton field in the configuration. We explore how the
relation between the hyperscaling violation parameter and the critical
dynamical exponent determine, in combination with the spacetime dimension, the
kinematic aspects of the fields. We provide a thorough study of the
thermodynamics including the quasi-local computation of charges, the
verification of the first law and arguments concerning the stability. Lastly,
we explore the effects in the thermodynamics from varying the rich parameter
space. We pay special attention in comparing the qualitative behavior of the
thermodynamics scalar-free solutions and the ones with a nontrivial dialton.
