Here (see the file) is the discussion of the case you are interested in. It is referred as "nearly degenerate perturbation theory", and it is reduced to a usual degenerate perturbation theory by a simple rearrangement of terms in Hamiltonian. The resulting secular equation differs from the degenerate perturbation theory by a simple modification:
\begin{equation}
\begin{pmatrix}
E_m^{(0)} + V_{mm} - \epsilon_\alpha & V_{mn}\\
V_{nm} & E_n^{(0)} + V_{nn} - \epsilon_\alpha
\end{pmatrix}
\begin{pmatrix}
c_m^\alpha\\
c_n^\alpha
\end{pmatrix} = 0.
\end{equation}
where $c^{\alpha}_k$ are the coefficients of the zero-order eigenstates
$|\alpha\rangle = c^\alpha_m|\psi^{(0)}_m\rangle + c^\alpha_n|\psi_n^{(0)}\rangle$ with the first-order corrected energies $\epsilon_\alpha$.
The rearrangement is performed as follows. Consider a Hamiltonian $H = H_0 + V$, where
\begin{equation}
H_0 = \sum_i E_i^{(0)}|\psi_i^{(0)}\rangle\langle \psi_i^{(0)}|,
\end{equation}
\begin{equation}
V = \sum_{ij} V_{ij}|\psi_i^{(0)}\rangle\langle \psi_j^{(0)}|,
\end{equation}
where $E_n^{(0)} \approx E_m^{(0)}$ for some $m$,$n$. Now, let me write \begin{equation}
H = H_0' + V',
\end{equation}
where
\begin{equation}
H_0' = H_0 - \frac{E_m^{(0)} - E_n^{(0)}}{2}(|\psi_m^{(0)}\rangle\langle \psi_m^{(0)}| - |\psi_n^{(0)}\rangle\langle \psi_n^{(0)}|),
\end{equation}
\begin{equation}
V' = V + \frac{E_m^{(0)} - E_n^{(0)}}{2}(|\psi_m^{(0)}\rangle\langle \psi_m^{(0)}| - |\psi_n^{(0)}\rangle\langle \psi_n^{(0)}|).
\end{equation}
The Hamiltonian $H'$ has two strictly degenerate eigenstates $|\psi_m^{(0)}\rangle$ and $|\psi_n^{(0)}\rangle$ with the energies $(E_m^{(0)} + E_n^{(0)})/2$, which allows to use usual degenerate perturbation theory by $V'$.
