  An edge (vertex) cut $X$ of $G$ is $r$-essential if $G-X$ has two components
each of which has at least $r$ edges. A graph $G$ is $r$-essentially
$k$-edge-connected (resp. $k$-connected) if it has no $r$-essential edge (resp.
vertex) cuts of size less than $k$. If $r=1$, we simply call it essential.
Recently, Lai and Li proved that every $m$-edge-connected essentially
$h$-edge-connected graph contains $k$ edge-disjoint spanning trees, where
$k,m,h$ are positive integers such that $k+1\le m\le 2k-1$ and $h\ge
\frac{m^2}{m-k}-2$. In this paper, we show that every $m$-edge-connected and
$2$-essentially $h$-edge-connected graph that is not a $K_5$ or a fat-triangle
with multiplicity less than $k$ has $k$ edge-disjoint spanning trees, where
$k+1\le m\le 2k-1$ and $$h\ge f(m,k)=\begin{cases}
2m+k-4+\frac{k(2k-1)}{2m-2k-1}, & m< k+\frac{1+\sqrt{8k+1}}{4}, \\
m+3k-4+\frac{k^2}{m-k}, & m\ge k+\frac{1+\sqrt{8k+1}}{4}. \end{cases}$$
Extending Zhan's result, we also prove that every 3-edge-connected essentially
5-edge-connected and $2$-essentially 8-edge-connected graph has two
edge-disjoint spanning trees. As an application, this gives a new sufficient
condition for Hamilton-connectedness of line graphs. In 2012, Kaiser and
Vr\'ana proved that every 5-connected line graph of minimum degree at least 6
is Hamilton-connected. We allow graphs to have minimum degree 5 and prove that
every 5-connected essentially 8-connected line graph is Hamilton-connected.
