The Jersey Archive does have several volumes of burial records, the same ones as the Soc J. There are records from all of the parish churches from the earliest records up until 1842.
There are St Helier burial records from Almorah and Mont a l'Abbe cemeteries plus memorial inscriptions from Green Street and St Peter's cemeteries. I believe St Clement burial records are on-line.
The Archive also holds an amazingly useful collection of funeral directors records. They are a reasonably new acquisition and are in the process of being indexed although many have already been done. These hold a wealth of information including date of death, address, place of burial and in some cases family members.
The Livres de Remarques from the funeral directors will give such information as burial number, name of person buried, relatives, age, date of death, cause of death etc etc - these are also at the Archive.