So your official contract has come to an end and your client is really happy with the work.  

Well done.  

But now they say they want you to "stick around for a bit" because one of their staff has quit and they're not sure how to proceed.  

What does "stick around for a bit mean"?

Remember, you're a professional.  You're an expert.  

You don't just hang around on the off-chance.  

Tell them you need to eat - so if you can't come to a financial arrangement, then you will be looking for something else.