  A path $P$ in an edge-colored graph is called \emph{a conflict-free path} if
there exists a color used on only one of the edges of $P$. An edge-colored
graph $G$ is called \emph{conflict-free connected} if for each pair of distinct
vertices of $G$ there is a conflict-free path in $G$ connecting them. The graph
$G$ is called \emph{strongly conflict-free connected }if for every pair of
vertices $u$ and $v$ of $G$ there exists a conflict-free path of length
$d_G(u,v)$ in $G$ connecting them. For a connected graph $G$, the \emph{strong
conflict-free connection number} of $G$, denoted by $\mathit{scfc}(G)$, is
defined as the smallest number of colors that are required in order to make $G$
strongly conflict-free connected. In this paper, we first show that if $G_t$ is
a connected graph with $m$ $(m\geq 2)$ edges and $t$ edge-disjoint triangles,
then $\mathit{scfc}(G_t)\leq m-2t$, and the equality holds if and only if
$G_t\cong S_{m,t}$. Then we characterize the graphs $G$ with $scfc(G)=k$ for
$k\in \{1,m-3,m-2,m-1,m\}$. In the end, we present a complete characterization
for the cubic graphs $G$ with $scfc(G)=2$.
