It's not too crazy. Start with 12 teams. One round of single elimination with the bottom 8 teams to eliminate 4. Then a typical 8-team double elimination, except no if-game for the championship. http://a.espncdn.com/photo/2015/0520/2015 SEC Tournament Bracket.pdf
The only way to get around that and finish the tournament in a single week is to limit the tournament to 8 teams and play a true double elimination. They are not going to do that because it would mean leaving 6 teams out of the tourney instead of just 2.