Connelly added that his exit, which opens up the seat for Republican Gov. Nikki Haley to appoint someone until a special election is held in 2014, is "going to open up a real ripple effect and we'll see a lot of candidates coming out" for the seat.

The South Carolina GOP is still figuring out the process, and what its role will be, Connelly said.

DeMint announced on Thursday he's resigning to head the Heritage Foundation, a conservative think tank based in Washington, D.C. An outspoken member of the Tea Party, DeMint said he was joining the foundation to help guide the Republican Party in the future.

"I've decided to join The Heritage Foundation at a time when the conservative movement needs strong leadership in the battle of ideas," DeMint said in a statement.