Representations induced from cuspidal and ladder representations of classical $p$-adic groups

Let $G_n$ denote either the group $Sp(2n,F)$ or $SO(2n+1,F)$ over a non-archimedean local field $F$. We determine the composition series of representations of $G_n$ induced from cuspidal and ladder representations such that the minimal exponent in the cuspidal support of the ladder representation is greater than or equal to $\frac{1}{2}$.