I was hired for an hourly job three weeks ago. I completed the job with the agreed upon hours. I have asked the client for a review, but he never responded. I did get paid for the work which was a section of the total work he wants, but the prior communication pre-contract suggested that I could finish the rest of the work if approved. I still haven't heard back, so I don't know if he wants me to continue or not. However, the contract is still open.
If you were asked to do tasks A, B and C...
...and you have only done task A....
Then you should continue until you finish all tasks.
This is an hourly contract. You will be paid for the time you log.
If you finish all requested tasks, then you should send one (only one) message asking about what to do next.
The client knows how to find you if he wants more work done.