Suppose we have a connected graph $H$ with $m$ edges and $n$ vertices, and we add an edge to it. How can one bound the number of spanning trees of $H \cup e$ in terms of $H$?

The following formula seems very plausible: if $\kappa(H) = \binom{m'}{n-1}$, then $\kappa(H \cup e) \leq \binom{m' + 1}{n-1}$.

In particular, this formula is easily seen to be true if $m' = n-1$ (the minimal possible value) and $m' = m$ (the maximal possible value).

Is there a quick reference or proof for this bound or something like it?

Thanks for the help