In some respects, I believe we can argue that the pigeon represents from where to where Stevens has to go. Remember the scene is when the pigeon enters and Stevens is startled by it, paralyzed with inaction- a similar theme in his time at Darlington Hall. His new owner is the one who approaches the bird and releases it. Stevens looks at it out the window, almost with a spirit of inspiration on his face. Perhaps, the meaning of this is that Stevens recognizes his folly in service, in silence, and now realizes that he, like the Post WW II world, has a new start on how to make things better. His recognition of it can represent an embracing of it.
I think the pigeon represents freedom.
the pigeon at the end of the film symbolises freedom, in opposition to Stevens' life, who is returning to serve at Darlington Hall.