My experience is in IT contracting for government, but here it is:
Old contract firm refused to even ask for a bill-rate increase on our tiny contract (2-3 positions) for 10 years. Finally they're going to ask for a $5/ hour bump, but to keep it all for themselves - not even a token increase for us. Instead we jumped ship to a small local firm on our next renewal. The old firm briefly, halfheartedly acted like they were going to put up a fight, but ultimately, old contract firm didn't even try and stop us.
Now, we had leverage - the 2 of us on the contract at the time were the only ones with up to date knowledge of the system we were supporting, and had done very good work for client for over two decades combined. Basically the client wanted US. They were willing to pay more for US. They did not care who the middle man was - the state has literally hundreds of contracting firms on a kind of collective contract that are willing to fill the middle-man's role in these arrangements. Therefore, we had a lot of leverage in that situation, and the middle-man had very little.
If you've got that kind of leverage, you can try and organize something like that for the next renewal. If not, ask for your raise at renewal (many months before renewal is the time to start asking), and be prepared to look for another position.