Melee +X
This creature gains +X Attack dice when it makes a melee attack. If the attack makes multiple attacks against the same or different objects, it gains this bonus only for the first attack it can make with this bonus. Does not grant a melee attack if this creature does not have one. Does not affect ranged attacks, damage barriers, or other non-melee attacks.
Multiple Attacks
Some attacks consist of multiple separate attacks against the same or different targets. Multiple attacks include those with the Doublestrike, Triplestrike, or Sweeping traits.
sIKE, there's no exception for the Hand of Bim-Shalla. The creature gets +1 melee until the end of the round, but +1 melee only adds an attack die to the first attack you can make with the bonus, within a given melee attack. A Hydra with Melee +1 from a Hand attacks at 4,3,3.
It's important to note, though, that if you use Battle Fury, the second attack is a completely separate attack sequence, and you will get the benefits of the Hand on your second attack action.