This will sound like a joke, but I'm for real. Just go to a hardware store, buy a cheap C-clamp and put it on your headstock (you might want to add something soft so it won't screw up your finish). That ought to tell you if the fat finger will work.
Sustain tricks and gadgets all have to do with adding mass to the headstock or bridge:
Hold your headstock against your speaker cab. Fat finger. I even remember these brass plates you used to be able to buy that were cut out in the exact shape of your headstock, with holes for the tuners, and you bolted them onto the back of your headstock.
The other options are magnetic gadgets like E-Bow and the Sustaniac that was so popular in the '80's. (I think Hernandes still makes it.)
My opinion? If you want lots of natural sustain, buy a big, fat, heavy, neck-through, mahogany guitar in the first place. You can change pickups and add all kinds of doo-dads to any given guitar, and they may have some effect, but usually a guitar just is what it is...
Fernandes make those sustaining guitars. The Edge of U2 used to use an EBow for 'with or without you' until he got wise to the fact that if you have the money, why hold something when the guitar can do it for you; so he bought 3 or 4 I think it is. Check them out at:
I have a Fat Finger on my Am Strat. It is not magnetic and simply adds some mass to the headstock. The amount of sustain it adds is very small but my Strat needed just that little bit to make it the way I wanted it. I had also removed the neck to clean the sawdust out of the neck pocket (only a tiny amount anyway) and checked that the butt of the neck was tight against the body. I also tightened the vibrato springs to pull the end of the bridge tight against the body (this required a complete adjustment of the set-up). These three things added up to small increase in sustain that I feel helped and finally made my Strat the way I thought it should be. Still, the difference is very, very small compared to strings, set-up, amp gain, playing style, etc. The right thing to do is to sell the guitar and replace it with one that sustains the way you want it. Only people who are obsessive and with nothing better to do (like me) care or act on these things. Aaron is right about the fatter neck and a fixed bridge might be the real way to go.