"Heir's heir" is probably the most succinct phrase since it very accurately describes what you are referring to. It looks a little awkward and if you keep following the chain of heirs it can get hard to track the various generations. But "heir" does not have a pattern similar to "[[great]grand]child" — there is no "[[great]grand]heir".
There are some slightly relevant terms that can make the description easier to understand:
order of succession — An order of succession is the sequence of those entitled to hold a high office such as head of state or an honour such as a title of nobility in the order in which they stand in line to it when it becomes vacated. This sequence may be regulated through descent or by statute.
successor — next occupant of position: somebody or something that follows another and takes up the same position
descendent — deriving or descending from an ancestor
One option, therefore, is to switch terms from "heir's heir" to something like "heir's successor." They are effectively synonyms but it does help readers understand exactly what you mean.
Phrases that do not work for various reasons:
second-in-line — Second-in-line is not necessarily the heir's heir. It often can be but it assumes that the if the current heir was suddenly not eligible (e.g., died) then that person's heir would become the current heir.
second heir — The second heir refers to the hierarchy of heirs on the same tier. The first heir could be the oldest male; the second heir could be the next oldest. This more accurately applies when the inheritance is not given to just one heir but is instead split across all heirs.