Keep in mind that the relations T = ∂E/∂S and 1/T = ∂S/∂E are normally associated with different levels of abstraction: T = ∂E/∂S is implied by the fundamental

Keep in mind that the relations T = ∂E/∂S and 1/T = ∂S/∂E are normally associated with different levels of abstraction: T = ∂E/∂S is implied by the fundamental relation of (phenomenological) thermodynamics, whereas 1/T = ∂S/∂E gets derived from a microscopic theory via statistical mechanics.