One of the nice features of the first admissible ordinal after $\omega$, i.e. $\omega_1^{CK}$, is that it is the collection of ordinals whose order type is that of a computable well-ordering on $\omega$.

Is a similar thing true for all other admissible sets? Specifically suppose $\alpha$ is an admissible ordinal, $\alpha^+$ is the next admissible after $\alpha$ and $\alpha \leq \gamma < \alpha^+$. Is there always an $\alpha$-computable well-ordering on $\alpha$ of order type $\gamma$?

If so can someone also provide a reference for the result?