Hi,
I'd just write up a letter of agreement that describes the project, the specific tasks you are/will be performing to complete the project (including the deliverables or outcome), and the associated fee and payment terms. You sign it like a regular letter, then under your signature, put
Agreed to:
with a line for your client's signature and the date.
In terms of the approach? Having an agreement is just good business. I would have had an agreement for the original "test" period. So, I'd just give the client the letter and say that "Since the project is continuing, I put together a letter of agreement to help both of us keep track of our work together" or something like that.
I am not an attorney, and this is not legal advice. You may want to consult an attorney for advice on this issue. I've found that if I manage the project correctly, a letter of agreement is all I need, but an attorney may have other advice.
Hope this helps.
Kathleen