Have you considered not raising your rates at all?
Rather, outsource maintenance work. Let's say, you're earning $50 per hour now. Find another freelancer to partner with, one who charges a little less than you do currently (there are some, myself included). Then, outsource the work to that person and make a small profit on their work.
Do some math: There are typically 40 hours in a work week.
If you spend 40 hours doing work that gets you paid $50 per hour, that's $2000.
If your outsourced help spends 40 hours per week doing work that gets YOU paid $50 per hour, that's $2000.
If you have to pay your outsourced help for 40 hours per week at $40, that's $1600 for them.
And $400 for you.
Without raising your rates for your clients and without having to turn away work. You will have to spend a few hours per week in management mode, though.
The problem in this is finding a competent professional to work with. You need to make sure you're not outsourcing work that is going to reflect badly on your business. You might outsource things like updates and backups, but not front end design work just yet. Make sure you have a good non-compete agreement and a non-disclosure agreement as well. Make sure you have someone you feel can and will communicate and work with you well. Make sure you treat that person well and fairly ... pay on time and be reasonable with deadlines.
I used to have a programmer that I worked with this way and it was a great way to work. His work was good and consistent and fast. I paid on time. We got along like a house on fire.