Yes a new bucket is cheap and if convenient, go get another one.
But, I also agree with one of the other replies, that it will be perfectly safe. Dish soap is designed to not leave a film and a bucket is hardly porous enough to be of concern. Rinse in hot water, then fill with hot water, add cap full or two of unscented bleach, leave for awhile then empty. If this was a bucket used to change oil in the car, then yeah, don't use it.
There are so many containers I have used that have gotten cleaned with dish soap afterwards and then used again and not a single issue. At some point equipment needs to be cleaned and sanitized. Hot water alone doesn't kill all bacteria, mold etc.
There are many posts here and on other forums where people use dish soap, acid, Mean Green cleaner and other stuff to clean old aquariums. And then rinse and sanitize with bleach. And silicon is a lot more porous than a plastic bucket.
Again, I'd either replace or use depending on how close you are to the store and whether you need an extra bucket for something else.
Remember Dawn is the official soap used to clean oil off birds, seals and other animals during oil spills. I know it gets rinsed off and compared to an entire lake or ocean that the amount of residual soap is nothing compared to this issue of the bucket. But still, there's never a mention of any concern that it could hurt fish.
While talking about Dawn, if you recently bought a bottle or will be, activate your dollar donation at:
http://wildlife.dawn-dish.com/en_US/savingwildlife/bcform.do
You don't have to give a dollar extra, just by buying the bottle of Dawn, Dawn will give $1 if you activate the code on your bottle.