Complete all 25 missions of Story Mode under any difficulty level to unlock Another Story Mode in which you assume the role of Laura Sevenbridge and her robot.
Select any mech in Story Mode
Finish Another Story Mode once. Now start another round of the normal Story Mode (New Game, not Continue), and you'll be able to select any mech that is also available on VS mode, except for the last three (General, Nextant 1 and Nextant 2).