More than likely the University would hire a search firm which is pretty customary in all sports, and industries generally speaking.
Not sure a search firm is needed though. We all know the hot position coaches, coordinators, smaller school HCs from hither and yon.
If you had to form some sort of committee with UK football ties, I think you go with relatively modern guys who have touch points in the NFL and or coaching circles.
Tim Couch-Former QB grapevine...Small community of guys who've played it in general, let alone at the SEC and pro level. Should have plenty of former teammates specifically offensive guys who are in the game to some degree coaching, commentating, working in athletic departments, scouting, etc...
Randal Cobb-Long NFL career. Generally, well liked. Knows a lot of people. Probably has former teammates coming up the ranks. Knows position coaches and coordinators.
Wesley Woodyard-Basically same as Couch and RC...7 degrees of separation theory. Worked with, played for, with, and against a lot of people.
Randy Sanders-Just retired, I think. His whole career is football country south of the Ohio River. He knows the terrain. Has a lot of former players, positions assistants, coordinators, contacts, who may be in the business to some degree if not rising up the ranks and worth a call to ask about so and so.
If not those exact people, then at least the criteria of Played, coached, NFL, P5 touch points.