For the higher terms, you can compute that there is an isomorphism \pi_*(E \widehat{\wedge} E^n) = E_*E \btimes ... \btimes E_*E, where \btimes is the L-complete (or in this case, I think, m-adically completed) tensor product. This is in my paper with Barthel,
arxiv.org/pdf/1410.5269.pdf, Cor. 1.24 (it's probably elsewhere in the literature as well), and in particular is also concentrated in even degrees