When running caste system you could also temporarily assign a few artists. This will cause the city borders to expand more quickly, granting faster access to all workeable tiles as well. Bit similar to a culture bomb really except you don't need a GA, although the effect's not immediate either.
I woudn't recommend running caste system for this purpose alone however, it's more of an added convenience for when you've adopted that civic already.
Another way of speeding up the regrowth process is to simply have a few missionaries and a batch of workers ready. Send in the missionaries right after capture so the city produces a fairly decent amount of culture right away (works best with No State Religion or Free Religion, obviously)
As soon as the borders expand, send in the workers to improve the tiles as you see fit. This is a good idea regardless because the AI is generally still less-than-optimal in its improvent choices, or at least in my opinion.
Be wary of horse archers, knights, cavalry and gunships though, the AI loves to snatch workers with those.
Finally, Hereditary Rule can be of some help as well: simply leave a few troops - typically damaged ones that need to heal anyway - in the newly conquered city for an instant happiness boost.