Perhaps the best time to inquire about anything at Greenville Memorial is in the spring or early summer. It will be closer to your graduation time. That way you can explore any available opportunities that might come across. Look at the Greenville Memorial Hospital website...try google to get to the website, I believe it is www.gmh.org
not sure, though. Hot jobs/positions that are hard to fill might pay a sign-in bonus. It is a big hospital and they use travel nurses, too.
Something else...try Spartanburg Regional, too. It is located in Spartanburg which is a big city 30 min. north of Greenville. Spartanburg Regional is a big hospital, too. Last time when I looked they had a lot of positions open for RN's. Also, I think that Spartanburg Regional was the first and only one hospital in the state to use the robotic system DaVinci for cardiothoracic surgeries.
Hope this helps. Good luck!