Let M be a hyperbolic 3-manifold such that ${partial}M$ has at least two boundary tori ${partial}_oM$ and ${partial}_1M$. Suppose that M contains an essential orientable surface P of genus $g$ with one outer boundary component ${partial}_oP$, lying in ${partial}_oM$ and having slope ${lambda}$ in ${partial}_oM$, and $p$ inner boundary components ${partial}_iP$, $i=1$, ${cdots}$, $p$, each having slope ${alpha}$ in ${partial}_1M$. Let ${eta}$ be a slope in ${partial}_1M$ and suppose that $M({eta})$ is toroidal. Let $hat{T}$ be a minimal essential torus in $M({eta})$, which means that $hat{T}$ is pierced a minimal number of times by the core of the ${eta}$-Dehn filling, among all essential tori in $M({eta})$. Let $T=hat{T}{cap}M$ and denote by $t$ the number of components of ${partial}T$. In this paper we prove: (i) if $t{geq}3$, then ${Delta}({alpha},{eta}){leq}6+frac{10g-5}{p}$, (ii) If $t=2$, then ${Delta}({alpha},{eta}){leq}13+frac{24g-12}{p}$, (iii) If $t=1$, then ${Delta}({alpha},{eta}){leq}1$.