  We study the stability of the Positive Mass Theorem (PMT) in the case where a
sequence of regions of manifolds with positive scalar curvature $U_T^i\subset
M_i^3$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF)
which may not be uniformly controlled near the boundary. Then if $\partial
U_T^i = \Sigma_0^i \cup \Sigma_T^i$, $m_H(\Sigma_T^i) \rightarrow 0$ and extra
technical conditions are satisfied we show that $U_T^i$ converges to a flat
annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.
