I'm not happy with the initial rate I accepted and am wondering if
it's appropriate to negotiate the rate moving forward before signing
the contract extension?
I assume that you aren't happy with the initial rate, but you took it anyway because you needed the job. And now that you completed the contract, you figure you can get more.
It's possible, and probably worth asking for a bit more in a professional way.
Be prepared with what you'll do if the reply is "No".
Many companies (like the one I work for) assume they can extend a contract pretty much indefinitely without renegotiating the rate. And in my company, if the rate needed to be changed, I'd have to write a new requisition, have it approved up several levels of management, and send it out to the recruiters to fill - even if the original contractor was ready to re-apply. I know this sounds foolish, but it is the way some big companies work. I can extend the period easily, but changing the rate is a big deal.
If the answer is "No" you want to be ready to either walk, or accept the current rate.