I just finished my first hourly paid job here and need some help on what to do next. The customer has been billed for the hours I worked last week (the money of which, according to my transaction history, are "in review") and I haven't sent the file to him yet. He mentioned he needed more of my services in the future. Do I have to end the contract now in order to get paid, or can I get paid and keep the contract open for future work? Thanks!
Ending an hourly contract has nothing to do with whether you get paid, or when you get paid.
The only thing that ending a contract does is prevent you from logging further time.
Ending a contract now WILL allow you to receive feedback from the client. That has nothing to do with payment.
If you don't have anything you can work on RIGHT NOW, then I suggest you tell the client:
If you do not need me to work on anything right now, then you are welcome to close the contract. That way there will be no confusion about whether or not you want me to continue working on this. There will be no way for me to bill any more time. And if you want me to work for you in the future, you can easily find me here on Upwork and create a new contract.
"But if you think you might want me to work for you some in the next few weeks, even if it's just small amounts of work or answering questions via email, then you can go ahead and leave the contract open and I'll be able to log time for that work."
As Preston pointed out, you don't need to have the contract closed in order to get paid on hourly contracts . Please, check this help article for more information about the Weekly Billing Cycle and communicate with your client regarding the contract status.