My hip flask has been ever so slightly flattened on the curved front. I seem to remember there being a little trick you can do to pop it back out again.

Anyone know what it is?

I got mine back to shape using a heat gun (for paint stripping).
Seal the top with the lid, then gently heat the flask from a distance until the air inside expands and pushes the dent out.Guess you could try a hairdryer too, but might not get hot enough.

Worked for my pewter flask, but its not got a leather or fabric cover.

Compressed air. Make / bodge a fitting and slowly increase pressure

Pop it in the freezer with the top open, fill the sink with hot water, remove from freezer, close the top and dunk in the hot water.