I wouldn't say there is a best time during the month to change hosts if I am honest. I would definitely say to change hosts before your next billing date though so that you do not end up paying for hosting you are no longer going to be using if you change.
I would say any time is the best time to change hosting, especially if you are having issues with your current hosting.
The best time to change hosts is around the Holidays, you can get discounts that way. Some hosting companies will have discounts regardless of the time of year if you make the switch to them. 😉