It's hard to diagnose a mechanical problem over the internet. Like does second gear make the noise when shifted into second or all the time in second gear? The person who wrote the article in Motorcycle Consumer News about putting the shift kit in his Yamaha YZF 600 (I know it's different from an R6) had the following to say about his bike before he put the kit in:
"Your left foot will encounter more crunching balkiness upshifting to second and downshifting to first on this bike than on any other current 600 (the article was published in 1/2001). False neutrals are also readily available in several nearby locations. The problem is most upsetting during high rpm clutchless upshifts during spirited sport rides or track days, but it's noticeable even when calmly tooling around town."
If that sounds like your problem, I would try installing the shift kit and hope that it cures your second gear noise.