Zefram Cochrane tried to kill himself in a deleted scene
In Star Trek: First Contact, Zefram Cochrane, portrayed by James Cromwell, was hesitant about making the first flight of the prototype warp ship, Phoenix. He had a problem with alcohol and wasn’t particularly interested in being known as the man who gave warp technology to mankind. He had no interest in being a hero, and according to a deleted scene, he didn’t have an interest in living.
In that scene in First Contact, Cochrane is chased by Starfleet officers from the Enterprise because he doesn’t want to make that first flight. He ends up being trapped at the edge of a cliff, but before he could jump, which would have killed him, Counselor Troi (Marina Sirtis) climbed up onto the cliff in an effort to convince him to come back down. Then she surprised him by telling him to jump and even helped him with a little push. But Geordi LaForge (Levar Burton) had already rigged up a force field that stopped Cochrane’s fall in mid-air.
According to co-writer Brannon Braga, the scene was cut for time, and they also felt like they had enough about Cochrane’s character so they removed the scene. (Star Trek Monthly issue 26, p. 35)
It didn’t make sense that Zefram Cochrane would be suicidal
The scene was actually the one Cromwell used for his audition for the role of Zefram Cochrane so he was sorry that the scene was deleted. But, after a discussion, he understood why. Had Cochrane really been suicidal, taking the first flight wouldn’t have been an issue for him. And that would have eradicated one of the film’s main story points which was Cochrane’s uncertainty about taking the flight and finally his agreeing to do so. (Star Trek: Communicator issue 111, p. 71)
Cochrane had enough going on in First Contact without including such a scene anyway. So, as the producers decided, including this scene simply wouldn’t have made sense.