I am sure that some of the other suggestions here probably are a lot simpler than what I am about to write but for what I make and how I work this is what I use for applying objects to rings.
I like to work on the flat. Then I extract some iso curves from the object I want to apply to and create an untrimmed four sided surface from those curves. From the surface I use _CreateUVCrv to give me the planar outline of that surface.
Using the curves from _CreateUVCrv I then run PlanarSrf. (Ooops still editing but pressed post)… Then I use _FlowAlongSrf with history on and tweak away.
Works for me but you might find one of the other ways easier.
In the case of this cyclinder I used I would have actually just put the seam to the back skipped extracting the Iso curves and then ran the rest of the process as above.