In Ina Paha, in an alternate reality that Steve McGarrett experienced as a result of being drugged, due to the Hawaii Five-0 Task Force not being created and presumably Jerry not eventually socializing with others, thereby improving his social skills, the Jerry of that reality subsequently became a very paranoid man with an undiagnosed mental illness as he was seen repeatedly talking to himself. He was also homeless, presumably suggesting that his mother had also probably kicked him out as she was no longer able to cope with him.
Jerry is a fan of Elvis Presley, which was discovered when Five-0 was called to investigate a murder of an Elvis impersonator in the episode. In the same episode, he shows off his musical talent by singing at the open mic when hanging out with the team for drinks after the case. 
In Season 7 episode 21, the Five-0 team gets into trouble while looking for a suspect on an uninhabited island. Jerry knows that they are in trouble, so he calls HPD to get support. After he saves the team, he finally gets his badge from Five-0.