For potential tails consisting of an inverse-square term and an additional attractive 1/r^m term, V(r)∼[ħ^2/(2M)][(γ/r^2)-(β^(m-2)/r^m)], we derive the near-threshold quantization rule n=n(E) which is related to the level density via ρ=dn/dE. For a weak inverse-square term, -1/4 < γ <3/4 (and m>2), the leading contributions to n(E) are n=^(E→0) A-B(-E)^sqrt[γ+1/4], so ρ has a singular contribution proportional to (-E)^(sqrt[γ+1/4]-1) near threshold. The constant B in the near-threshold quantization rule also determines the strength of the leading contribution to the transmission probability through the potential tail at small positive energies. For γ=0 we recover results derived previously for potential tails falling off faster than 1/r^2. The weak inverse-square tails bridge the gap between the more strongly repulsive tails, γ≥3/4, where n(E)=^(E→0)A+O(E) and ρ remains finite at threshold, and the strongly attractive tails, γ<-1/4, where n=^(E→0)Bln(-E/A), which corresponds to an infinite dipole series of bound states and connects to the behavior n=^(E→0)A+B E^((1/2)-(1/m)), describing infinite Rydberg-like series in potentials with longer-ranged attractive tails falling off as 1/r^m, 0<m<2. For γ=-1/4 (and m>2) we obtain n(E)=^(E→0)A+C/ln(-E/B), which remains finite at threshold.