After a horrible accident Jack a loving husband and father awakens in hospital with anmesia, sitting next to him is a woman who is not his wife but she convinces him otherwise.

That exactly why "Unknown" work, instead of an amnesic trying to recover his memory with external help, the hero never lost his memory despite everybody trying to convince him that he is crazy.
If he going to receive an inheritance, maybe they are trying to swindle him of his money with an actor taking his place. Maybe he read about the inheritance in the media, when doing research on himself.
But again to do research on himself he needs to know who he is.
The only way your amnesia would work is if he was the double of a rich/influential/important guy and that they uses him as a patsy for their scheme. He play the other guy when they rob, spy him and then kill him. However in the mean somebody that he loves is in danger and he wants no part of that scheme.
Maybe his daughter needs a kidney transplant and he had been kidnapped before the operation. Now he has to race across time to find his daughter and extirpate himself from the con.

>> he has an ID on him

Which enables them to contact his real wife and children. End of story.

(If the story is set anywhere near the present tense, it's set in a Big Brother world of electronic surveillance and massive aggregation of personal data such that it's now harder than ever to lose one's identity, to drop out of sight.)

I would assume he has an ID on him and he is told he in about to inherit.