Certainly the explicit renegotiation should be part of the new contract negotiations, prior to agreeing to the contract. However, you should take steps to ensure that both your request is not a surprise to them, and that you fully understand the financial terms of the employment arrangement.
In particular, if you were a contractor before and not an employee, do not be surprised if the employee rate is actually lower than the contractor (as the benefits/taxes may be higher for a full employee, even part time). (Note, this is true in the US, not sure about Canada.)
Either way, you should do what you can to ensure both you and your boss know what the other expects in an informal environment before you get to a formal negotiation, especially if that negotiation is with a HR recruiter rather than your direct boss. In addition to not surprising him/her, you also give your boss a chance to make sure the budget has room for your salary increase.
On your end, perhaps prior to talking to your boss, you should find out if there are other employees who have been similar situations in the past, and whether a salary increase was automatic, or even possible. (Your coworkers may not feel comfortable talking numbers, but likely they will at least be willing to tell you the possibility of an increase.) Given you're at a large company and in a common role there, it seems likely there should be some past history available to you to help guide your expectations, and to give you some ammunition (but don't name names unless they give you permission to do so).
