Can I make a bid for a client out of contract. Even if I have worked for the company that has current contract?

The answer to your question depends on the nature of your relationship with your current employer.  Do you have any sort of contract (including an oral one) with your current employer which limits your right to compete with it?  Are you using any sort of confidential information belonging to your employer (for example, is their pricing structure and/or customer list confidential) to try to poach the client?  If the answer to both of these questions is no, there should be no reason why you can't try to "steal" the client.  Of course, if you try and are not successful, you will almost certainly be fired.