Option 1: Rig it with it’s own bones. Then animate it. This can be apart of the same armature or a separate armature. Side Note: You can select multiple armature and switch to pose mode. This allows you to animate both without having to jump back an forth between object mode and select the different armatures.
Option 2: Is weight paint it so that the neck bone has some influence over the “Tail”
Option 3: Create a shape key. You can animate the shape key manually or setup a driver to a bone.
{Just a thought. Not sure if it would actually work} Option 4: Assign tail to vertex group and use that along with a shrink wrap modifier. Although it might work better using bones with a shrink wrap constraint. Weight paint to the bones. Then adjust influence. Again this is just an idea that popped into my head. Not sure how well it would work without testing it.
