We investigate first-passage percolation on the lattice $\mathbb{Z}^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $\tau$. We only assume $\mathbb{P}[\tau=0] < p_c(\mathbb{Z}^d)$, the critical probability threshold for Bernoulli bond percolation on $\mathbb{Z}^d$. We obtain lower bounds of order $n^{-\alpha_d}$ (where $\alpha_d > 0$ 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 $\tau$, 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.