In this paper we study the two-body elastic scattering process in ϕ3 theory, in the limit s-->∞ with t equal to or near the threshold value 4, where the mass of the scalar particle is taken to be unity. For the scattering amplitude corresponding to any ladder-type diagram, we find that there is a promotion for the power of s by 1/2 . For example, the scattering amplitude for the ladder diagram of n+1 rungs is promoted from O(s-1lnns) away from threshold to O(s-12lnn-1s) at threshold. This means that, for small coupling constants, the leading Regge pole is promoted from the neighborhood of l=-1 away from threshold to the neighborhood of l=-12 at threshold. There are, in addition, infinitely many Regge trajectories approaching l=-12 as t approaches 4. The scattering amplitude for the n-rung diagram is explicitly given in the limit s-->∞, with t at or near the threshold 4, and various scales for t-4 are pointed out. The two-rung and three-rung diagrams are especially studied in detail.