We present the calculations of head-to-head domain wall (DW) structures in magnetic nanowires of permalloy and cobalt with a tapered cross section, with average widths, $W_{\mathrm {mid}}$ , ranging from 60 to 300 nm, ratios of the bottom width, $W_{\mathrm {Bottom}}$ , to top width, $W_{\mathrm {Top}}$ , ranging from 1 to 2 and thicknesses, $T$ , from 7.5 to 15 nm. In agreement with prior work, for wires with a rectangular cross section, the critical thickness and wire width $W$ which determines whether transverse or vortex DWs are energetically favored is approximated by $T\cdot W=C\cdot {\delta }^{2}$ with $\delta $ the exchange length and $C$ a constant. When taking the slope of the tapered edge of the nanowires into account, the relation is modified by a slope-dependent term which is presented in this paper. The results provide an extended phase diagram for stable DW regimes. Furthermore, the effect of magnetocrystalline anisotropy energy is investigated which yields a more complex thickness and anisotropy energy-dependent behavior.