This is one more question on flat morphisms that I have been thinking about.
Suppose $f:X\rightarrow S$ is a flat projective morphism of noetherian schemes. Both $X$ and $S$ are smooth and irreducible. Let $T$ be a smooth closed subscheme of $S$.
Suppose $F$ is a coherent $O_X$ module such that $F_T$ is flat over $T$.
Is there some additional condition which will ensure that $F$ is $S$-flat?