What's the difference between "fresh out of sth" and "out of sth"?

Example:
We're out of milk.
We're fresh out of milk.

Question

What's the difference between "fresh out of sth" and "out of sth"?

Example:
We're out of milk.
We're fresh out of milk.

Answer

Fresh out is used to express it happened very recently.

I’m out = Doesn’t say when you ran out
I’m fresh out = implies you recently ran out.

Answer

1) fresh out of college - have recently left, more applicable for places
2) we ran out of gas - no longer have it, more applicable for items

Answer

As others have suggested, 'fresh out' can have an idea of being recent, but it doesn't have to. It can just be a colorful way to say that you don't have anything at all. (such as the Clint Eastwood quote)

A. 'What's our next step?'
B. 'I dunno. I'm fresh out of ideas.'

Answer

Eric is correct.
"Fresh out" means that it just happened, perhaps even a few minutes ago.

Sometimes people will just say the shortened version:
"Do you have any lemons?"
"We're fresh out."

Answer

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