  In this paper, we consider the asymptotic behavior of two Teichm\"uller
geodesic rays determined by Jenkins-Strebel differentials, and we obtain a
generalization of a theorem in \cite{Amano14}. We also consider the infimum of
the asymptotic distance in shifting base points of the rays along the
geodesics. We show that the infimum is represented by two quantities. One is
the detour metric between the end points of the rays on the Gardiner-Masur
boundary of the Teichm\"uller space, and the other is the Teichm\"uller
distance between the end points of the rays on the augmented Teichm\"uller
space.
