In his Master's thesis, Ján Mazák proved that the circular chromatic index of the type 1 generalized Blanuša snark $B^1_n$ equals $3+{2\over n}$. This result provided the first infinite set of values of the circular chromatic index of snarks. In this paper we show the type 2 generalized Blanuša snark $B^2_n$ has circular chromatic index $3+{1/\lfloor{1+3n/2}\rfloor}$. In particular, this proves that all numbers $3+1/n$ with $n\ge 2$ are realized as the circular chromatic index of a snark. For $n=1,2$ our proof is computer-assisted.