This paper is a continuation and a supplement to the previous paper [3]. For any element x of a *-regular semigroup (with *-operation*) in the sense of K. S. S. Nambooripad and F. J. C. M. Pastijn [1], there exists a unique inverse x^+ of x such that x^*Hx^+. The main purpose of this paper is to investigate several conditions for a *-regular semigroup S to be a special *-regular semigroup with respect to the unary operation + : S→S.