Seasoning a carbon steel wok
It sounds to me like you are using too much heat to allow the seasoning process to work, instead the oil is burning off before it can polymerize and form the seasoning layer.

The bits that are already seasoned probably don't need to be re-seasoned; you can add seasoning to the areas where it is not seasoned. You may want to do multiple layers to ensure that the coatings are even across the whole pan.

I suspect that what you need to do, as you have been rubbing thin layers of oil over the surface, but then use a lower heat setting and ensure that the pan is fully heated through over as much of the pan as you can get hot and at constant temperature. One way to do this is to heat it in an oven, rather than over a flame.