First, let's look at terminology (which is important when you've learnt a particular phrasing of the laws). A time-varying magnetic field is when a magnetic field at a point is changing in time. A space-varying magnetic field is when the magnetic field is varying in space. And that means a static magnet is sufficient to produce a space-varying magnetic field; a moving magnet produces a time-varying magnetic field (and yes, it's space-varying too).
Now obviously, the moving magnet produces a circulating electric field around it: if you draw any loop in space, and the magnet goes through it, the flux through the loop changes in time, producing an emf and so forth, according to Faraday's law.
So, to answer your question, if you move a magnet simply in space, there is an electric field that goes around it, following it as it goes. It's almost like the magnetic field if you send a short localized "pulse" of positive, immediately followed by negative, current moving through a wire (which you can't though - without a complicated charge removal mechanism - and we're ignoring other induction effects).
Perhaps you meant to ask what happens when the magnet is stationary, and the conducting rod goes through the field. We'll look at it from the rest frame of the magnet. At first, there is no electric field - it is the magnetic Lorentz force itself (which emphatically does not require a space-varying magnetic field) that is causing the emf in the rod, and moving the charges to the ends. There is equilibrium when the electric field created due to this polarization of the rod creates an equal and opposite emf. But you've said this much already.
The interesting question to ask here is: what is causing the polarization in the rest frame of the conducting rod, where $v = 0$ and there is no magnetic Lorentz force? The answer is that the magnet is moving in this frame, producing a time-varying magnetic field. And that means the electric field is back, and this polarizes the rod, until the charges compensate with an opposing electric field.
In summary, an electric field, when not caused by charges, is caused only by a time varying magnetic field (a space varying magnetic field is a magnetostatic field, and produces no electric field of its own). And the presence of an electric field might depend on your frame of reference (something that is explained better by special relativity).