Hospital spokeswoman Olena Scarboro said that Joseph Asher of Greenville is in critical condition at the Joseph M. Still Burn Center in Augusta, Ga.
Asher had been listed as a missing person before he walked into a Greer hospital nude and covered with burns.
He was later transferred to the Georgia center.
Multiple media outlets report that the 38-year-old man climbed an electrical transformer assembly, likely grabbing equipment that had 100,000 volts of electricity running through it.
The remnants of the man's clothing were found at the site. Authorities have not said why he was chasing the rabbit.