In Hall's Marriage Theorem, we have a set $B$ of brides and $G$ of
grooms, where each bride $b$ has an acceptable set $A_b \subseteq G$
of grooms. A **matching** $m:B\to G$ is an injection such that $m(b)
\in A_b$ for each $b\in B$. (So each bride gets married, but some
grooms may be out of luck.)
Obviously, if there's some set $S \subseteq B$ of brides
such that $|\cup_{b\in S} A_b| < |S|$, then a matching is impossible;
the theorem is that this is the *only* obstruction.

I'm interested in the case that $G$ is an interval $[1,n]$ in $\mathbb N$, and each $A_b$ is a subinterval $[i,j]$. (Perhaps each bride is only willing to accept grooms within a certain range of heights.) I have been able to make the following refinement: if no matching is possible, then there is an interval $[x,y]$ such that $|[x,y] \cap G| < |\{b : A_b \subseteq [x,y]\}|$.

(This is an improvement in two ways -- it restricts the form of $S$, plus the left side is a priori larger than $|\cup_{b\in S} A_b|$.)

Is this refinement known?

This extension wasn't very difficult, but if it was known I'd rather give credit to its earlier discoverers.