The rules are not precise on this question. I would say you have three options, depending on how much of a penalty you want to apply:
EASY -- The new adventurer has the full amount of available actions
MEDIUM -- The new adventurer has the same amount of available actions as the newly deceased adventurer had left over when he/she died
HARD -- The new adventurer doesn't appear at the Tomb Entrance until the start of the next round.
I suspect the normal way to play is the "MEDIUM" option, but I suggest you use whichever option feels the best to you.
Also, don't forget to advance the Comet Track once when a character dies.
Great suggestions... That probably needs to go into the Errata. (if not already there)