Conjecture points to Hurst Head House being close to Hurst Hall and being at the top - the 'head' - of Hurst.
It was consistently inhabited throughout the 19th century and beyond. Hurst Head House is probably the older of the two adjoined buildings and is mistakenly, I think, now labelled as Hurst Hall on some modern maps and the listed buildings register.
Here is a list of relations who have lived or stayed at Hurst Head at some time or other.