Find inner product of two states given inner product of an orthogonal state

The answer depends on the dimension $d$ of the Hilbert space.

If $d = 2$ then $|phirangle = frac{e^{itheta}}{sqrt{2}} (|irangle - |jrangle)$ for some $theta in [0, 2pi)$ and so $langle i | phi rangle = frac{e^{itheta}}{sqrt{2}}$. In other words, it can be any complex number of absolute value $frac{1}{sqrt{2}}$.

If $d > 2$ then $|phirangle = frac{a}{sqrt{2}} (|irangle - |jrangle) + b|krangle$ for some $a, b in mathbb{C}$ such that $a^2 + b^2 = 1$ and any $|krangle$ orthogonal to $|irangle$ and $|jrangle$. Therefore, $langle i | phi rangle = frac{a}{sqrt{2}}$. In other words, it can be any complex number whose absolute value is in $[0, frac{1}{sqrt{2}}]$.

(I assume that $langle i|j rangle = 0$ and all kets are normalized.)