  Let $p \geq 5$ be a prime, $N$ be an integer not divisible by $p$,
$\bar\rho_0$ be a reducible, odd and semi-simple representation of
$G_{\mathbb{Q},Np}$ of dimension $2$ and $\{\ell_1,\cdots,\ell_r\}$ be a set of
primes not dividing $Np$. After assuming that a certain Selmer group has
dimension at most $1$, we find sufficient conditions for the existence of a
cuspidal eigenform $f$ of level $N\prod_{i=1}^{r}\ell_i$ and appropriate weight
lifting $\bar\rho_0$ such that $f$ is new at every $\ell_i$. Moreover, suppose
$p \mid \ell_{i_0}+1$ for some $1 \leq i_0 \leq r$. Then, after assuming that a
certain Selmer group vanishes, we find sufficient conditions for the existence
of a cuspidal eigenform of level $N\ell_{i_0}^2 \prod_{j \neq i_0} \ell_j$ and
appropriate weight which is new at every $\ell_i$ and which lifts $\bar\rho_0$.
As a consequence, we prove a conjecture of Billerey--Menares in many cases.
