Borrowing liberally from SkewT python 3 port, I added a height scale to a plot with the code below (the class is just a holder). Most would be easy to port into Metpy.SkewT with some finagling.

Two obvious problems would need to be worked out:

The axis for height doesn't show up. I've tried a number of settings with no luck.
Better(smart?) positioning of the axes and wind barbs.

from metpy.units import units

class Sondes(object):
    def __init__(self, sounding, dt):
        self.file = sounding
        self.dt = dt
        self.dt_str = dt.strftime('%Y-%m-%d %HZ')

        self.p = sounding.variables['pressure'][:]
        self.T = sounding.variables['temperature'][:]
        self.Td = sounding.variables['dewpoint'][:]
        self.u = sounding.variables['u_wind'][:]
        self.v = sounding.variables['v_wind'][:]

        # Calculate the parcel profile.
        self.H = mpcalc.pressure_to_height_std(self.p)
        self.zero_levs = mpcalc.find_intersections(self.H, self.T, np.zeros_like(self.T))

    def plot(self, fig=None, xlims=None, ylims=None, plot_barbs=False, bold_temp=None, add_height=False):
        if fig is None:
            fig = plt.figure(figsize=(7, 7))
        skew = SkewT(fig)
        _ = skew.ax.set_title(self.dt_str)
        _ = skew.ax.set_ylim(ylims)

        skew.plot(self.p, self.T, 'r', linewidth=2)
        skew.plot(self.p, self.Td, 'g', linewidth=2)

        if bold_temp is not None:
            for temp in bold_temp:
                skew.ax.axvline(temp, color='c', linestyle='--', linewidth=2)

        if plot_barbs:
            _ = skew.plot_barbs(self.p, self.u, self.v)

        # This order of plotting matters
        _ = skew.ax.set_xlim(xlims)
            
        if add_height:
            majorLocatorKM = MultipleLocator(1)
            majorLocatorKFT = MultipleLocator(5)
            minorLocator = MultipleLocator(1)

            plims = skew.ax.get_ylim() * units('hectopascal')
            zmin = mpcalc.pressure_to_height_std(plims[0]).magnitude
            zmax = mpcalc.pressure_to_height_std(plims[1]).magnitude
            zminf = zmin * 3.2808
            zmaxf = zmax * 3.2808

            kmhax = fig.add_axes([0.775, 0.1, 1e-6, 0.8], frameon=True)
            kmhax.xaxis.set_ticks([], [])
            kmhax.spines['left'].set_color('k')
            kmhax.spines['right'].set_visible(False)
            kmhax.tick_params(axis='y', colors='k', labelsize=8)
            kmhax.set_ylim(zmin, zmax)
            kmhax.set_title("km/kft", fontsize=10)
            kmhax.get_yaxis().set_tick_params(which="both", direction='out')
            kmhax.yaxis.set_major_locator(majorLocatorKM)
            kmhax.yaxis.set_minor_locator(minorLocator)

            fthax = kmhax.twinx()
            fthax.xaxis.set_ticks([], [])
            fthax.tick_params(axis='y', colors='k', labelsize=8)
            fthax.set_ylim(zminf, zmaxf)
            fthax.get_yaxis().set_tick_params(which="both", direction='out')
            fthax.yaxis.set_major_locator(majorLocatorKFT)
            fthax.yaxis.set_minor_locator(minorLocator)
        return skew

Height scale for SkewT #262

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