The other answers describe a rainbow-like phenomenon involving ice crystals, which may very well be what you saw. However, there is another possibility.
A normal rainbow occurs when light enters a spherical drop, refracts at the curved surface (dispersing the colors), reflects off the back of the drop, and then leaves the drop, refracting again. The angle between incident and outward-going light is about $42^\circ$, and so you see rainbows $42^\circ$ from the point directly opposite the Sun.
No one said, though, that the light had to undergo exactly one total internal reflection before leaving the drop. It can reflect multiple times, coming out at different angles. A second-order rainbow is at a slightly different angle from the normal first-order one. More interestingly, third- and fourth-order rainbows can be found circling the Sun (not circling the point opposite the Sun), simply due to the geometry. Wikipedia has some information, though unfortunately I cannot find any good diagrams for this effect online. Third-order rainbows are very hard to see, but they have been documented and are in a sense more "true" to the definition than phenomena involving ice.