I have had a car that takes 0W-20 since 2011 (or well, actually, I have had two cars). At the dealership, they sometimes put 5W-30 oil there. I don't understand why since the manufacturer of the car has specified 0W-20, and not 5W-30.
My experience is that 5W-30 when put to a car that takes 0W-20 oil does not cause any engine damage. Also, I used to have a really old car that took 5W-30. Well, sometimes some oil change guys put 5W-50, thinking that since it's an old car, let's put some really thick oil in. Even that 20-unit difference caused no damage.
SAE 20 means 5.6 - 9.3 cSt and SAE 30 means 9.3 - 12.5 cSt viscosity @ 100 degrees Celsius. So, as you can see, a thick SAE 20 can actually be equal to a thin SAE 30.
The 0W vs 5W means the 5W oil will be harder to start in the extreme cold (such as -20 degrees Celsius). The -30 part is more worrying, meaning its viscosity is larger than -20 oil. It causes more fuel consumption. In theory, the additional viscosity could affect how the oil flows around the engine.
I wouldn't change the oil immediately after putting 5W-30. I would just keep on driving with the 5W-30 and then at the next oil change, use the correct grade (0W-20).