Order type of $\alpha$-computable well-orderings

Surprisingly, no there isn't! See also this paper.

Roughly, if $\alpha$ is sufficiently stable, then the next admissible is much greater than the supremum of the $\alpha$-computable well-orderings of $\alpha$.

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Specifically, the relevant fact is:

If $R$ is an illfounded binary relation on $\omega_1$ in $L_{\omega_1}$, then there is an infinite $R$-descending sequence $\sigma$ with $\sigma\in L_{\omega_1}$.

Quick Lowenheim-Skolem arguments show that if $\alpha$ is "sufficiently like" $\omega_1$ (I think $++$-stable is enough), the same is true: if $R$ is an illfounded binary relation on $\alpha$ in $L_{\alpha}$, then there is an infinite $R$-descending sequence $\sigma$ with $\sigma\in L_\alpha$. It's now not hard to show that for such an $\alpha$, the supremum of the order types of $\alpha$-computable well-orderings of $\alpha$ (I'm denoting this by "$\omega_1^G(\alpha)$" in a paper I'm writing - I don't think there's standard notation) is not admissible (hence is $<\alpha^+$).