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I am not a lawyer. But in school I did have a class entitled "engineering and the law" taught jointly by a lawyer and an engineering prof. They said that if there is a contract, the owner is as specified in the contract (normally the client). If the contract doesn't address the issue, the copyright resides with the author. If the client paid the author, there is an implied permission to use the work in the context it was written for. The upshot would be that the author (in the absence of other contract provisions) can reuse the work in other contexts, cannot deprive the client of use in the intended context, and the client cannot use it in another context. As I understood it then.
~~As others have said, you had best speak with a lawyer.~~

I am not a lawyer. But in school I did have a class entitled "engineering and the law" taught jointly by a lawyer and an engineering prof. They said that if there is a contract, the owner is as specified in the contract (normally the client). If the contract doesn't address the issue, the copyright resides with the author. If the client paid the author, there is an implied permission to use the work in the context it was written for. The upshot would be that the author (in the absence of other contract provisions) can reuse the work in other contexts, cannot deprive the client of use in the intended context, and the client cannot use it in another context. As I understood it then.
As others have said, you had best speak with a lawyer.