  We minimally coupled a scalar potential $V(\phi)$ with asymmetric vacua to
the Einstein gravity to numerically construct the hairy Reissner-Nordstrom
black hole (RNBH) as a direct generalization of RNBHs to possess scalar hair.
By fixing the electric charge to mass ratio $q$, a branch of hairy RNBHs
bifurcates from the RNBH when the scalar field $\phi_H$ is non-trivial at the
horizon. The values of $q$ are bounded for $0 \leq q \leq 1$, which contrast to
a class of hairy black holes with $q>1$ in the Einstein-Maxwell-scalar theory.
We find that the profiles of solutions affected by the competition between the
strength of $\phi_H$ and $q$, for instance, the gradient of scalar field at the
horizon can increase very sharply when $q \rightarrow 1$ and $\phi_H$ is small
but its gradient can be very small which independent of $q$ when $\phi_H$ is
large. Furthermore, the weak energy condition of hairy RNBHs, particularly at
the horizon can be satisfied when $q>0$.
