Your prestige currently (C+) is lower than Radford's (B-) and probably a lower C+ than Southern's. You're likely qualified for all jobs at a C prestige and below, right (and "keep looking" for a portion of the C+ jobs)? When you first started, Radford was a D, but they've had good results and have moved up to B-, so that comparison isn't really fair.
The job logic just doesn't seem to let you jump up in prestige with a move.