Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice available. Now can this be affected by how much the cardinality of $ON$ is far from that of $V$? 

I mean suppose we axiomatize that $V=H_{<ON}$ and that $|X| \not \leq |ON| \leftrightarrow |X|=|V|$, then this says that the cardinality of the class of all set ordinals is immediately below that of the class of all sets, and that there are no other cardinalities other than those of sets, so the cardinalities of $V$ and $ON$ are very near.

On the other hand suppose instead of the above biconditional we stated that there are $|V|$ many proper class cardinalities$^\dagger$ other than $ON$ and $V$, then this would entail that the cardinality of the class of all set ordinals is quite distant from that of $V$ if not extremely lower.

>Would those have different choice properties, that in some sense form a spectrum intermediate between set choice and global choice? For example, the former entailing more amount of choice than the latter?

$^\dagger$ this can be captured by saying that there is a class that is the disjoint union of pairwise non-equinumerous proper classes such that the domain class of that union is equinumerous with $V$. If we further demand that any two of those proper classes have comparable cardinalities, then $ON$ would be hugely lower in cardinality than $V$.