  In this paper we continue our analysis of arXiv:1308.1478[hep-th] and study
in detail the parameter space of three families of doubly spinning black ring
solutions: balanced black ring, unbalanced ring and dipole-charged balanced
black rings. In all these three families the Extremal Vanishing Horizon (EVH)
ring appears in the vanishing limit of the dimensionful parameter of the
solution which measures the ring size. We study the near horizon limit of the
EVH black rings and for all three cases we find a (pinching orbifold) AdS$_3$
throat with the AdS$_3$ radius $\ell^2=8 G_5 M/(3\pi)$ where $M$ is the ring
mass and $G_5$ is the 5d Newton constant. We also discuss the near horizon
limit of near-EVH black rings and show that the AdS$_3$ factor is replaced with
a generic BTZ black hole. We use these results to extend the EVH/CFT
correspondence for black rings, a 2d CFT dual to near-EVH black rings.
