Can a toroidal transformer with two secondaries, inner and outer, simultaneously induce current in both?
The magnetic field powers both secondaries.


  ...since almost all of the magnetic flux is contained within the primary coil...


The magnetic flux is contained in the magnetic core, not in the primary coil.

Any loop of wire that links the core will have a voltage induced in it by changes in the core flux. That is, all loops of wire that link the core have exactly the same voltage induced in them. It doesn't matter what order these are put onto the core, they still get the same voltage (to first order, any differences due to leakage inductance will be vanishingly small, parts per ten thousand in typical iron-cored transformers)