My contract is up, but the client wants me to "stick around for a bit".

What you gonna do?

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.