You didn't say what phase the job was in. If there is a significant amount of uninspected work, I would call for a "partial" inspection on the work done before you put your name on the permit. The inspector may be able to shed some light on what is going on if there were problems.
I would go the partial inspection route, and take lots of pictures to document what shape the job was in when you took over. As far as ethics, I do not see a problem, unless you are friends with the other EC. You are assuming responsibility for any work that is already covered up, if you know the other ECs work is good than you have no worries.
Life is tough, Life is tougher when you are stupid