Never Say You're a Consultant
Most people don't care what your employment status is. As long as you're doing your work and helping others do theirs, they don't care if your an old-timer, new employee, consultant, or intern. So unless asked, don't make your status known. Instead, behave like an employee and you're more likely to be treated like one.
During one contract at a bank, I did such a good job of behaving like an employee that I was nominated for an award. Of course, I wasn't eligible for the award since I was a consultant. However, because the process was initiated, my manager learned of the nomination which, in my mind, is as good an outcome as I could've hoped for. It's always good to have your manager know that the people you're working with are happy with your work. Especially when it comes time to renew a contract or negotiate a rate increase.