There's a standard definition of the $\alpha$-jump of $\emptyset$ ($\emptyset^{(\alpha)}$) for recursive $\alpha$ (I don't know why I've never seen an $\alpha$-jump of a set $X\neq\emptyset$ before for $\alpha>\omega$.) A concept that appears a lot in the context of ordinal notations in recursion theory is a "copy", which is a set (usually of lower rank) equipped with an ordering that's isomorphic to some other set. For example, an ordinal notation with order type $\varepsilon_0$ is a copy of $(\varepsilon_0,<)$.
However, even though the The supremum of $\alpha$-recursive relations need not be the next admissible after $\alpha$ (mentioned in MO question #277343), such an $\alpha$ where this happens is called non-Gandy. However, Noah Schweber hasmentions a good relevant statementsuccessful characterization of admissibility in this answer: the countable admissibles are exactly the ordinals with no copy that's computed by a copy of any smaller ordinal.
There's a standard definition of the $\alpha$-jump of $\emptyset$, or "$\emptyset^{(\alpha)}$", for recursive $\alpha$, this usually comes along with a collection of definitions for the "hyperarithmetical hierarchy". The reason this is well-defined in spite of there being multiple ordinal notations for $\alpha$ is Spector's theorem that the resulting sets are Turing equivalent regardless of notation chosen. The $\alpha$-jump may be generalized from $\emptyset$ to an arbitrary set of naturals as well, but I am not sure if Spector's theorem applies, we may instead need to define $0^{(e)}$ for ordinal notation $e$.