Most cows I have seen have the hair on the outside so that shouldn't be a problem, Planning on doing a back quiver with goatskin hair on next. What sort of quiver are you looking at building? I found looking at pics then trial and error with card and paper templates was the only way to really get what I wanted.
Theres plenty of quiver pics on here and other forums and websites on the net for inspiration.
Cheers,
Mike