We investigate first-passage percolation on the lattice $\Z^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $τ$. We only assume $¶[τ=0]0$ is explicit) for the probability of having two disjoint geodesics between two pairs of neighbouring vertices at distance $n$. Additionally, under more specific assumptions on the distribution of $τ$, we obtain similar lower bounds for the probability of having two disjoint geodesics (except for their starting and ending points) between the same two vertices.