The coil hasn't got any fields of any sort if it's just a coil of wire- picture a spring sitting on a worktop, it's not exactly an electromagnet is it.
When a magnet is moved through it, the magnet has a field, which moves with it. This means a magnetic field moves through the coil, and so there is a changing magnetic field (at one moment there's none, next there is some, so a change must have occurred) So you get an emf.