In this paper we study first-passage percolation in the configuration model with empirical degree distribution that follows a power-law with exponent $τ\in (2,3)$. We assign independent and identically distributed (i.i.d.)\ weights to the edges of the graph. We investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical distances. When the underlying age-dependent branching process approximating the local neighborhoods of vertices is found to produce infinitely many individuals in finite time -- called explosive branching process -- Baroni, Hofstad and the second author showed that typical distances converge in distribution to a bounded random variable. The order of magnitude of typical distances remained open for the $τ\in (2,3)$ case when the underlying branching process is not explosive. We close this gap by determining the first order of magnitude of typical distances in this regime for arbitrary, not necessary continuous edge-weight distributions that produce a non-explosive age-dependent branching process with infinite mean power-law offspring distributions. This sequence tends to infinity with the amount of vertices, and, by choosing an appropriate weight distribution, can be tuned to be any growing function that is $O(\log\log n)$, where $n$ is the number of vertices in the graph. We show that the result remains valid for the the erased configuration model as well, where we delete loops and any second and further edges between two vertices.