I think it depends on what you think a big time coach is. Being in the SEC or BIG is a massive advantage nowadays. If you look at the ACC and Big12, there might be someone out there. Here are probably the 3 best that UK could land:
Matt Campbell is having a good year at Iowa State and has very strong Ohio connections, but he’s had some very mediocre years at ISU.
Chris Klieman is having a pretty good 3rd year in a row at Kansas State. Crushed it at the FCS level with 4 national championships in 5 years and has gone 10-4, 9-4 and currently 7-1 in the last 3 years.
Rhett Lashlee at SMU is in his 3rd year and has gone 7-6, 11-3 and currently 7-1 in his 3 years there. Offensive guy for those that want that side of the ball.