The best thing for you to do is go to the local union hall (IBEW) for the area you will be living in. Talk with the business manager there and he should let you know what it will take to join. The union rates are not to good in South Carolina -
IBEW Local 379. If you go the non-union route (dog eat dog, don't talk to your fellow workers about what you get paid, no holiday pay, no health insurance + supply your own drills, benders etc)
You can see i am a bit biased towards the union.
There will be a local council office which will have the requirements to become a registered/licenced electrician. I would say that in order to become an electrician in South Carolina, the best bet would be to contact all of the electrical contractors in the area, try and get an interview, explain your qualifications and experience and hopefully they will offer you a job. Never, ever, talk about the UK system being better than theirs. Remember, the guys you will be working with, believe that the USA are the best in the world at everything. Just try and blend in and learn as much as you can about the US system, study for an exam based on the current NEC (their equivalent of our IEE Regs)
South Carolina Electrical Contractor License/Exam | South Carolina After doing all of this you will have a greater respect for the US system and actually find a lot of their ways are better eg. the conduit bender.
The first stop should be at the local Union Hall and bring a fully paid up UK electricians union card with you and explain to them that you are a good union brother (the "International" part of the "I" BEW). They might have you work for a non - union contractor and get you to SALT for them. You can read up about that somewhere else.
Finding any work in that area right now is tough. Try and stay away from domestic installations though because it will be difficult for you to get any commercial work after that.
You are a highly skilled Union brother from the UK, who has a great respect for the American system and willingness to learn and develop your skills further.