Why not a $coNP$ hierarchy?
By definition, for any oracle for a language $A$, $(coNP)^{A}=\{L \mid \overline{L} \in NP^{A}\}$. Using this definition and definition of co-class we have 

$$L \in co(NP^{NP})\iff \overline{L}\in NP^{NP} \iff L \in (coNP)^{NP}$$

You can also use $coNP$ instead of $NP$ as oracle (as Yuval commented), since if we can decide $x \in L$ using an oracle for a language $A$ we can decide $x \in L$ using another oracle for $\overline{A}$ as well, because $x \in A \iff x \notin \overline{A}$.