In this paper we consider composition operators Cφ on the Hilbert Hardy space over the unit disc, induced by analytic selfmaps φ. We use the fact that the operator C∗φCφ is asymptotically Toeplitz to obtain information on the essential spectrum and spectrum of Cϕ, which we are able to describe in select cases (including the case of some hypercyclic composition operators or that of composition operators with the property that the asymptotic symbol of C∗φCφ is constant a.e.). One of our tools is the Nikodym derivative of the pull-back measure induced by φ. An alternative formula for the essential norm of a composition operator (valid in select cases), in terms of the aforementioned Nikodym derivative, is established. Estimates of the spectra of adjoints of composition operators are obtained. Based on them, we describe the spectrum of composition operators induced by maps fixing a point, whose iterates exhibit a strong form of attractiveness to that point.