does this allow a Hobgoblin to make 2 save. An initial save and then an immediate one?
Of course, for the progressive effects (minor effect initially, worse effect upon failing one or more saves, such as Sleep or Medusa turn-to-stone), the bonus save can actually make things WORSE, as you fall to the worse stages more quickly. So there may be times when they would be smarter to NOT use their ability.
Personally, I don't ever allow a character to be worse off because he was given an extra attempt to save. In the examples you give, I wouldn't have a character (or NPC) worsen until they failed a normal save. The alternative is that often extra saves will have no actual effect on the final outcome, just the speed at which the situation progresses (for better or worse).

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.