Your question is confusing.

1. As you said, if you have H1 visa (not status), then you must be able to re-enter without problem. The visa issued in the US is as much valid as the on issued in Korea

2. If you mean you have valid H1 status, but not visa on your passport, then you have to have a visa to have a international trip unless the destination is Canada (not sure about Mexico as of lot of illegal immegration issues around there)

You can get the visa without going out of US, but it takes time (2 – 3 months)

Good Luck

trip님의 글

I am working with H1 visa that was issued in US. I have not been in Korea to issue H1 visa at my country.

Under this condition, is there a way to go abroad on business (except Korea) and come back to US

without no need to visit Korea to get a H1 visa issued?