An edge-coloring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Lehner, Pilśniak, and Stawiski proved that all connected regular graphs except $K_2$ admit an asymmetric edge-coloring with three colors. We generalize this result for graphs whose minimal degree $δ$ and the maximal degree $Δ$ satisfy $δ\geq Δ/2$.