Make it cheaper and more popular than it was. This is highly unlikely. Or be a cult hit that's still a cult hit decades later with a cult that's grown.
The days of letter campaigns "saving" shows and getting them an additional season (e.g., Star Trek (TOS) or Remington Steele are long gone. And generally when they did succeed, you only got one more season. Shows have natural lifespans; they jump the shark, and oft-times, getting more episodes doesn't mean you actually get more of the show to love.
Something like Gilmore Girls: A Year in the Life or the recent revival of The X-Files really only happens because the episode count is low enough to make the budget relatively small, and they had found new audiences. Something like Longmire can continue on Netflix after having been cancelled by A&E, because it's a short-run show with a low budget and has good enough ratings for Netflix to find it desirable.
But it also had the advantage of still being in production. Once the cast moves on to other projects, and the sets are broken down and all the costumes and props go away, it becomes very tough to put it all back together again.
Shows don't get cancelled because they have huge mass audience appeal. Today's bazillion channel-mit-streaming outlets world, nobody gets a 40 share any more, like they did when there were only three broadcast channels. Everybody's clawing at smaller and smaller pieces of the pie, which is why we see so many shows with lower episode counts these days. Smaller risk for the networks to pull in the advertising dollar or subscription fees to pay for it.
As someone who lived through the Fox Friday-night-slot-o'-death-show-parade throughout the '80s and '90s, I will also note that SF shows tend to be higher on the kill list, simply because FX and building a world you can't just find adds to the budget. Fox was all set to renew James Cameron's Dark Angel for its fourth season, when they got a last-minute pitch from Joss Whedon, and they cancelled Dark Angel so they could pay for Firefly. Them's the breaks.
Personally, as a fan, the best thing you can do for yourself is participate in whatever activities that happen around the death of the show, just so you can mourn it, if for no other reason. But if nothing comes of it, give it up, let it go, and follow the showrunner to their next project.
The showrunner is the guy you follow. In the case of Fringe, that would probably mean keeping an eye out to see if J.H. Wyman is making any pilots for the 2017-2018 season* (or maybe hunting up Almost Human). Was I disappointed when Leverage went down? A bit, but I knew John Rogers was working on The Librarians and The Player. Was I crushed when Dead Like Me, Wonderfalls, Pushing Daisies, and Hannibal were cancelled? Of course. But I trundled after Bryan Fuller wherever he led (including The Amazing Screw-On Head, Mockingbird Lane, and High Moon, all of which went to pilot and then died), and I'm still following Bryan Fuller to American Gods. Was I heartbroken when Person of Interest was cancelled? Yes. But I let it go, and followed Jonathan Nolan over to Westworld. As a fan, that is what I do; be a fan of the creator.
- footnote added in 2021: J.H. Wyman has a new NBC show, premiering Mar. 1, Debris.