The Sixth Doctor first wore the blue coat for the webcast story Real Time. His normal multicoloured coat would have been very problematic to produce properly for the webcast.
Big Finish Audios, who collaborated in the production of Real Time, then decided to sue the blue coat as the Sixth Doctor's outfit of choice for a variety of stories. The CD covers of numerous Sixth Doctor audios show him wearing the blue coat, while the dialogue makes it clear that he's also wearing the blue coat.
However, we had already seen the Sixth Doctor's regeneration in Time and the Rani, where he's wearing the multicoloured coat. So, in a later Big Finish audio, the Sixth Doctor changes back to the multicoloured coat.
All these stories were only made from 2002 onwards, and in-universe they all take place between The Trial of a Time Lord and Time and the Rani.