Geoffrey Monk, from Banbury in Oxfordshire, drove a red Toyota Rav 4 across the grass and over the 550 foot cliff at around 4.30pm on Bank Holiday Monday.
The 77-year-old's vehicle landed upside down in the sea, just metres away from Eastbourne Cliff Rescue Team, which was trying to recover another body.
The first man found under the cliffs, who has not yet been identified, is believed to be aged between 20 and 30 and may have drowned rather than fallen to his death. It is thought he had been in the water for some days.
Coroner's officer Robin Emery is appealing for information and has released a description of the man's clothing.
He was wearing size seven white Fila trainers with velcro fastenings, dark jeans and a navy zip-up fleece with stripy navy and white sleeves. The man was between 5ft 8in and 6ft tall.
Anyone who thinks they may have known this man is asked to call Mr Emery on 414007.