There are many components all connected to keep the nose gear in line. If one component is worn or out of tolerance it can cause shimmy problem. Reference Dave McFarlane’s article: Can You Stop Nose Gear Shimmy instructions and suggestions.
Overlooked items also include the rod ends, shimmy dampener mounting and attachment, shimmy dampener, and steering collar. The steering collar is where the steering rod tubes connect and the upper torque link is attached to, along with the shimmy dampener on most aircraft. If the steering collar has play vertically and laterally and is allowed to tilt; that will cause excessive wear and force on the torque links, steering, and strut components. There are three different thicknesses of shims to help get the collar in place.
Ensure that the shimmy dampener is working properly with no dead spots in dampening action. The cylinder could be worn on the inside or on a piston that warrants replacement of components. All of the nose strut components are tied to each other and any movement is transmitted through to the tire and back through the dampening system, which if remedied will continue to be more pronounced.