Left the oil cap off after an oil change

Long story short, my dad was showing me how to change my oil for the first time and the oil cap got left on the engine cover after we finished. I drove off normally and didn't notice anything until I stopped at a gas station and saw smoke coming out of my engine. Opened up the hood and oil was everywhere, the cap was nowhere to be found.

I checked the oil and it wasn't dangerously low but we filled it back up and got a cap that barely fits and zip tied it on there for the time being. The check engine light is on, which I think is just due to the cap not being on, but I'm wondering if this caused damage or if there's anything I should do to prevent any. We already ordered a new cap and we'll have the engine bay detailed next week, but I'll still need to drive the car before then to get to work. It's still burning off some oil 2 days later but it's not leaking and I've been checking it religiously.

TIA