Honor the price you gave until the job is done and paid for. Any time after that, but before you have both consented to schedule another service is a good time. Say, “Ok I’m happy to schedule this job again, but I have to increase the price to $X. Was there a day/date in particular that works best for you?”
You don’t have to give them any more explanation than that. Plenty of times they will go for it without much of a fuss if your work was good the first time around.
Only if they prod you for an explanation should you bother telling them why you need more money.
Before you go through with the increase, ask yourself: Am I ok with losing this account completely?
Because that too-low price may be the very reason you have this account in the first place. Rocking the boat with a price hike may sink it. But there’s only one way to find out.
If your clients try to squirm and wheedle their way out a price increase when you have already stated the new rate, then my personal belief is that you do not back down. If they don’t agree to your new price then you let them go.
If you state an increase, and get bullied out of it, then they own you and that’s not how a business runs.