As Helix_FR stated the cylinder may have end shake. That means the cylinder moves in the frame too much forward and back. There are specs for that and a common fix is a washer on the axle of the cylinder that takes out the end shake by forcing the cylinder back a few thousandths of an inch.
My guess is the gun is ready for more than a few shim washers. It probably needs the factory to inspect and rebuild/adjust it. Usually when there is too much end shake there is also too big of a cylinder to barrel gap and you get lead spitting out of the cylinder gap which would now be excessive. But S&W would know as would a Revolver Certified S&W repair shop. These specs are basic to revolver repair as is the adjustment. The end result may be that the barrel has to be set back as well as the shim washers added.
S&W revolvers are fine guys but eventually can shot loose. That's the price you pay for finesse verses heavy duty (Ruger is the example of the later). But any S&W revolver that still looks good and shoots relatively good is worth repairing, in my opinion. But get it done right.
For a lot more information see the Kuhnhausen book on S&W revolvers: