Jesse James' Severed Hand was first seen in Sam & Max Hit the Road, where it was found on display in the Hall of Oddities. Sam pockets the jar and takes it to a Pickle Jar Opener specialist (in other words, a Snucky's soda jerk). With the hand free, it is used with the broken golf ball retriever to get Shuv-Oohl's mood ring which was stuck in one of the many layers of the World's Largest Ball of Twine.
Sam and Max apparently kept the hand as a trophy, as it reappears above the closet door in the office in the Telltale episodes, along with a plaque supposedly autographed by Jesse James. Sam has to occasionally clip its nails as they are constantly growing. In Abe Lincoln Must Die!, Sam and Max analyzed it using a portable carbon dater. This dated it as fourteen years old (in other words, dating back roughly to the time Hit the Road took place), revealing it to be an obvious fake. Max accepts this just fine, but Sam is in denial. It is unknown where they got the signed plaque that accompanies it.
Interestingly, while the hand in Hit the Road is a left hand, the hand in the Telltale episodes is a right hand. Also, the hand lacks half the ring finger from Reality 2.0 onwards, while it was complete up until Abe Lincoln Must Die!. No reason is given for this.
In Night of the Raving Dead, it becomes reanimated, holds up the diner, and takes Girl Stinky hostage. After a small shoot off with Sam, it gets trapped in some Gooey Molasses Tar Cake. Its sharpshooter skills made it the ideal hand attachment when altering Jurgen's Monster.