According to the Lorentz transformations of the electric and magnetic fields, there is also a magnetic field in a frame of reference moving relative to another frame of reference in which there is just a static electric field.
So, assuming an isolated point charge, in the frame of reference in which the charge is at rest, there is only a static, radially directed electric field.
But, in a relatively moving frame of reference, there is also a magnetic field.
Put another way, it isn't that we apply the Lorentz transformation to the electron and then calculate the field, we apply the Lorentz transformation to the electromagnetic field over all space.
The case of a neutral wire with steady current is a special case with a symmetry that allows us to derive some results without directly applying the Lorentz transformation to the fields themselves.
In summary, to understand the answer to your question fully, you must appreciate that the electric and magnetic fields themselves transform according to the Lorentz transformations - relatively moving observers 'see' different electric and magnetic fields.
Since you are reading Griffiths' "Introduction to Electrodynamics", please see Example 12.15 (of the 4th edition) Magnetic field of a point charge in uniform motion where Griffiths derives the magnetic field by transforming from the static electric field of the point charge's rest frame into a relatively moving frame.