  We extend the study of corrected thermodynamics for the 3D black holes
conformally coupled to scalar field up to non-perturbative level. We calculate
the exponential correction to entropy arises due to the microstate counting for
quantum states on the boundary. This exponential correction in entropy
attributes to the other thermodynamical quantities also. We study the stability
and phase transition for this system of black hole under the influence of
non-perturbative correction. We also discuss the quantum work associated with
exponential corrected entropy. Finally, we justify the results from the view
point of thermodynamic geometry.
