Border crossing Wadi Araba Israel to Jordan

I appreciate a sure answer please
A traveller entered Jordan at Wadi Araba border crossing coming from Elat.
He had entry stamp on a sheet of paper and was exempted from visa fees.
As i under stand this person must return to same border crossng to exit Jordan.
Is there a way to cross to Israel via other border crossing ie Sheik Husein Bridge in the north or possible leave via Amman airport
thanks in advance