"The very minute after"

Very is acting here in a sense closer to its old meaning of truly, precisely, literally:

You’re the very man they’re looking for!

The very fact that we’re alive is a miracle.

The given sentence is very similar to:

Literally the minute after I leave, they start taking my books from the shelf…

The evolution that very underwent, from this older sense to its more common modern usage as an intensifier, is the very same shift in meaning that literally is now undergoing, to the chagrin of many pedants.

When very is used in that context, it takes the place of mere, the form originally derived from the Latin merus (meaning pure or undiluted), a comparitive meaning "nothing less than" or "downright", or indeed "just", "only", "no more than".

In that context it particularly means immediately or "as soon as". It is pretty much always prefixed with "the".

The very idea of it.

The very best of luck.

That is the very worst thing that could have happened.