= Solution
Define the ingoing coordinate
$$
v=t+r_*,
\qquad
\frac{dr_*}{dr}=\frac1{f(r)}.
$$
Then $dt=dv-dr/f$ and
$$
\boxed{ds^2=-f(r)dv^2+2\,dv\,dr+r^2d\Omega_{d-2}^2}.
$$
This is the higher-dimensional analogue of <Ingoing Eddington-Finkelstein coordinates>. Its metric and inverse are regular at $r=r_+$, including when the zero is degenerate. A gauge transformation removes the singular radial term in the transformed potential, leaving the regular representative $A=-Q r^{-(d-3)}dv$. These expressions analytically extend the spacetime through the future outer horizon.
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