I felt that I could share something that has helped me after having tailbone pain from childbirth. My pain continued for over 18 months until I finally took up a swimming exercise which involved swimming with a snorkel and mask in the free style position. While swimming I focus on the kick up pushing upwards but then rather than pushing the legs downwards I let each respective leg fall down on its own. It is important to keep your legs straight and to flex your feet rather than point them and also to swim in a relaxed fashion. When I have had relapses of tailbone pain this swimming has helped.
If swimming is too strenuous, I have also just held on to a float and then kicked my legs up and down in this same fashion. I really hope this can provide some relief for others.