I've actually been considering writing a FanFic of Braska's adventure. I have a few ideas for it (including a plot-twist involving Gilgamesh. Read the Spoiler for more details).

Spoiler:
After Jecht attacks the Shoopuf, the trio are walking down the Djose Highroad towards the temple. Auron is telling Jecht that he's "an irresponsible drunk, and that nothing he could do can change his opinion". Jecht then says he could change, and would help everyone he saw from now on. It's at that moment that they see a wounded traveller (Gilgamesh) lying on the side of the road. Jecht immediately helps him up and takes him to Djose Temple. There, Gilgamesh explains that he travels throughout many worlds and dimensions in search of powerful swords. He came out in Spira because he sensed a Masamune was there. As a way of saying thanks for helping him, Gilgamesh gives Jecht the black sword you normally see him with. Liking the sword, Jecht gives Auron the Longsword and says "if I don't make it back to Zanarkand, give this to Tidus.