Is it permissible for a woman to marry a man who is already married to another, knowing that the man will not state in the contract that he is married?

There is no objection to a woman marrying a man who is already married, whether he has consummated the marriage with his wife or is only contracted to her. If she is pleased with his religiosity and character, there is no impediment. He is not obligated to inform her family that he is contracted to someone other than their daughter.