You understandibly only sync (a maximum of) 2 levels to JIRA: either 1st and 3rd or 2nd and 3rd.
I can use the variables on map-level to sync the missing level: fill JIRA label with {activity} or {task} (depending on the 1st setting).
However, they are only sent to JIRA once, on initial push.
When you later either...
a) move the card in storiesonboard, so that it falls under a different parent
or
b) change the name of the labelized card (activity or task) in storiesonboard
... this is not reflected in JIRA.
I can imagine you would not want to apply b on labels of JIRA issues that are already 'done' (although JIRA will show such a change in history, so it would be traceable, this might be a user preference / setting).

A "re-sync all previously synced items" would also do the trick, I guess. This might even work for some other stuff you can set on the "Issue tracker integration" page of the "board settings", like "add card link"-tick, "sync releases"-tick and all other JIRA fields configured here.
Or just "re-sync all" on save of that screen?