  By considering a small correction to the Maxwell field, we show that the
resultant black hole solutions (also known as the asymptotically
Reissner--Nordstr\"{o}m black holes) undergo the reentrant phase transition and
can have a novel phase behavior. We also show that such a small nonlinear
correction of the Reissner--Nordstr\"{o}m black holes has high effects on the
phase structure of the solutions. It leads to a new classification in the
canonical ensemble of extended phase space providing the values of the
nonlinearity parameter $\alpha$ being $\alpha \lesseqqgtr 4q^{2}/7$. We shall
study these three classes and investigate deviations from those of the standard
Reissner--Nordstr\"{o}m solutions. Interestingly, we find that there is the
reentrant phase transition for $\alpha <4q^{2}/7$, and for the case of $\alpha
=4q^{2}/7$ there is no phase transition below (at) the critical point. For the
last case, one finds that small and large black holes are thermodynamically
distinguishable for temperatures and pressures higher than the critical ones.
