I would imagine a candidate would have to file to run in either the general election or the special election (which will occur on the same day.) Mississippi has a primary for its general election in June, but the special is a jungle primary in November with no party affiliations listed.
I know Wrangor has posted about Chris Daniels before, who is a tea party type that is challenging the Senator appointed by the governor. That could split the vote and leave a path for a Dem to get a runoff spot (if no one gets above 50% of the vote, there would be a runoff later in November.) But it looks like there are currently two Dems running, so they'll split the vote as well. One of the Dems was the Secretary of Agriculture under Clinton, and wiki has this amazing pic of him from the time: