Current paths can be broken down into a superposition of equivalents.
Above the PCB, you have pins, then the lead-ins, then the windings themselves.
The windings are where almost all your wire length is, so it should be pretty obvious that that is your problem area!
Roughly speaking, stray inductance is proportional to wire length, period. You can always do worse, but you can never do better than the length equivalent.
Indeed, suppose the windings were short circuits: now you can see (and measure) the inductance which is due only to the lead-ins and pins. This is equivalent to a loop of wire following whatever paths are shown above. And, for a winding of that size, these cannot be more than about 20nH.
(This figure comes from the permeability of free space, \$\mu_0 = 1.257\,\mu\textrm{H m}^{-1}\$. The length you multiply by is not exactly the length of the wire: there's a geometry constant missing in there. But for a given shape, if you scale it up or down, its inductance will be proportional to its length dimension.)
So what is it about current paths, anyway? What about the path between primary and secondary? Ah, now that's the key. Get the primary and secondary wires as close as possible!
Wind single layers, alternating between primary and secondary. Use twisted pair (where possible). Use transmission line transformer design!
Indeed, understanding that the energy is not so much stored in the core, or its airgap, but simply the volume around the primary, is fundamental. With that understood, you will intuitively understand that the secondary must be, as much as possible, in the same place as the primary wire is.
This regards equivalent inductance only, which is only half the story about energy stored in that air gap between conductors -- but it's the first step towards understanding the whole.
Tim