No, that's incorrect. A hash function should have these properties:

* Given a hash h it should be hard to find a message m such that hash(m) = h

* Given a mesage m1 it should be hard to find m2 such that hash(m1) = hash(m2)

* It should be hard to find any two messages m1 and m2 such that hash(m1) = hash(m2)

If you have some secure hash function h, the function h'(m) that appends the low byte of m to h(m) is still secure under all three properties, but it obviously leaks information.

Hmm, I unfortunately can't remember the hash attack models right now, but you may be right. I think there are models under which this fails, though.