Morrison always denied the charge but died before his appeal was heard
Mr Crist will leave office in January and fans of The Doors hope he will make good on a promise to review the Morrison case before he stands down.
Morrison was convicted of drunkenly exposing himself on stage at a concert in Miami, Florida in March 1969. He denied the offence and was appealing his sentence - a six month jail term and $500 fine - when he died of a heart attack in July 1971, aged 27.
Fans first petitioned Mr Crist in 2007 to look again at the "trumped up" charges and consider a posthumous pardon. Mr Crist agreed but made little progress.
With only two months left in office, he has promised to revisit the case.
"Candidly, it's something that I haven't given a lot of thought to, but it's something I'm willing to look into in the time that I have left. Anything is possible," Mr Crist told Washington blog The Hill, adding that fans should "stay tuned".
Dave Diamond, a Doors fan from Ohio, sent the original petition to Mr Crist and has run a letter-writing campaign in the intervening years.
He said: "This is the last chance Gov Crist will have with the board to make good on his 2007 commitment to give this longstanding case a fair review and a vote in favour for or against the special posthumous pardon request.
"We are still hopeful that he will see this case through. He's the only governor in Florida that's even acknowledged this effort at all, and we appreciate his time towards this matter."
Several audience members back up Morrison's claim to have simulated indecent exposure rather than doing the actual deed. Unfortunately, none of the photographs taken on the night capture the incident.
Pardons in Florida must go through the Board of Executive Clemency, which meets on December 9.
There is a precedent: in 2003, former New York Governor George Pataki pardoned the late comedian Lenny Bruce for a 1965 obscenity conviction. Bruce had peppered a stand-up show in New York with more than 100 "obscene" words. Undercover detectives were in the audience and testified against him.