  Introducing a new form of scalar potential $V(\phi)$, we derive a proper form
of the rotating black hole solution in three-dimensional Einstein gravity with
nonminimally coupled scalar field and find that the first law of thermodynamics
of this new rotating hairy black hole can be protected, where the scalar field
parameter $B$ is constrained to relate to the black hole size. We also disclose
the Hawking-Page phase transition between this rotating hairy black holes and
the pure thermal radiation. Moreover, we study phase transitions between this
rotating hairy black hole and rotating BTZ black hole. Considering the
matchings for the temperature and angular momentum, we find that the rotating
BTZ black hole always has smaller free energy which is a thermodynamically more
preferred phase. Additionally, we evaluate the thermodynamics of the rotating
black hole with minimally coupled scalar hair in three dimensions, which
exhibits that the thermodynamical behaviors of this rotating hairy black hole
are very similar to those of the rotating black hole with nonminimally coupled
scalar hair.
