We report on bcc-like phases in ultrathin Fe films grown by thermal deposition on $\mathrm{Cu}(111)$ previously thought to consist exclusively of fcc phases distinguished only by their magnetic order. Our scanning tunneling microscopy and spectroscopy data together with published x-ray photoelectron diffraction results [M. T. Kief and W. F. Egelhoff, Jr., Phys. Rev. B 47, 10785 (1993)] provide us with sufficient detail to deduce the film structure. Two growth regimes are considered: (1) films with 1--2 monolayer average thickness grown near 200 K, which nucleate as bcc-like bilayer islands: Larger islands show bcc-like fringes coexisiting with an fcc center domain; i.e., the bcc-like phase is stable only within a certain distance to a step edge. The presence of a bcc-like bilayer phase provides a straightforward explanation for the ferromagnetism previously observed in these films. In addition we find that the bcc-like phase can be promoted by H adsorption at $80\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The bcc domains form ``displacement vortex'' structures to simultaneously minimize film stress and interface energy. (2) In films grown at room temperature, between pseudomorphic fcc areas, we observe a more ideal but still strained bcc phase in regions with a local thickness of at least 4 monolayers. Also in this growth regime, the fcc-bcc transformation is facilitated by step edges, which are abundant due to the imperfect layer-by-layer growth.