My understanding of the particular topics (involved in the question) is quite incomplete (both conceptually and w.r.t. knowledge), so there might be some mistakes in the question.
Suppose we have some reasonably powerful model of infinitary computation (both with (i) transfinite time and (ii) transfinite tape or variables). Also both the maximum running time and maximum variable values both go sufficiently beyond $\omega{_C}{_K}$ (to avoid any trivial issue). We also assume that the program starts from all variables equal to $0$ or blank tapes etc.
The exact time for which a program runs might slightly depend on convention (it seems). But nevertheless, under very reasonable conventions, for every ordinal $\alpha \in \omega{_C}{_K}$ there must exist some program that halts (reaches the halt state or line-number that is) at a time "exactly" equal to $\alpha+1$.
Now suppose we want to halt at a point beyond $\omega{_C}{_K}$. A simple method that comes to mind is the following:
" We simply start from $0$ and start enumerating all the pairs of the form ($\alpha$, some member of $\mathcal {O}$ corresponding to $\alpha$) in increasing order of $\alpha$. Note that first element in these ordered pairs is an ordinal and second is just some element that belongs to the set $\mathcal {O}\subset \mathbb{N}$.
If we are given all such ordered-pairs for some some ordinal $\alpha$ it is trivial to calculate them for $\alpha+1$. For some limit element $\beta$ we just have to do little more work. We list all the indexes corresponding to (ordinary) programs computing total recursive functions. Then we see for which of the indexes $n \in \mathbb{N}$ we have:
$$ | \phi_n(0)| \,, |\phi_n(1)| \,,|\phi_n(2)|\,,|\phi_n(3)|,...$$
forming a fundamental sequence for $\beta$. This should be easy to check. And with this we can get all pairs of the form ($\beta$, some member of $\mathcal {O}$ corresponding to $\beta$). But when we have $\beta=\omega{_C}{_K}$ none of the indexes should succeed. At this point we know we have reached beyond $\omega{_C}{_K}$ and the program terminates. "
However, my question is that with this method when we reach the time of $\omega{_C}{_K}$ for running time of the program, it will take us some further time to ascertain that there is no element of $\mathcal {O}$ corresponding to $\omega{_C}{_K}$. And hence the program will reach the halt state at a time well-beyond $\omega{_C}{_K}+1$. Can we devise a method that will make the program reach halt state exactly at $\omega{_C}{_K}+1$ (or smallest time while being $\ge \omega{_C}{_K}$)?