There's not really such a thing as "gaining unauthorized access to the salt" when you already have the hash; the salt is just as secret as the hash, and the hash is useless as a means of authentication without the salt, so obtaining the hash, unauthorized or not, generally also means you obtain the salt. Libraries for algorithms like scrypt even usually give you one string which contains both a random salt and a hash.

You can regenerate a rainbow table which uses that salt, but you'd have to generate a rainbow table for every password, since each password has its own random salt. I don't know how rainbow tables work exactly, but I'd assume an old fashioned brute force attack or dictionary attack is cheaper than making a rainbow table for each password.