We prove high-dimensional asymptotics for the time constants in first-passage percolation (FPP) on Z^d along all diagonal-like directions v=(1, 1, .., 1, 0, 0, ..., 0) of f(d) nonzero entries. We show that if f(d) ~ o(d), the time constant along the direction v behaves similarly to the axis directions, whereas if f(d) is a linear fraction of d, the asymptotic behavior is characterized by the Lambert W function. The proof was based on a cluster exploration idea, which allowed us to estimate moments of non-backtracking first-passage times as well as to fix an error in [AT'16]. This talk is based on joint work with A. Auffinger, C. Guo, W.-K. Lam and T.-L. Lu.