To keep hydrangeas, or other woody plants, from "suckering," you need to remove as much of the woody material as you can. If you have already cut the bush down at ground level, you'll just need to remove any roots you can — the deeper you dig, the better.
The second option is herbicides. Because you have just cut the hydrangea down, with no leaves to apply the herbicide to, you'll have to use the method called "stump application." Use an herbicide that contains the active ingredient Triclopyr and paint it on exposed fresh cuts on the woody surfaces with a paint brush or some other applicator. Be sure to follow label directions, including the interval to wait before replanting. Using herbicide is not guaranteed to kill right away. There's a chance that the stumps may sprout for some time and need further treatment.
