Filter Design for Phase Response
It's instructive to see what an ideal filter adding a constant phase shift would look like. If $\theta$ is the desired phase shift, the corresponding ideal frequency response is

$$H(e^{j\omega})=\begin{cases}e^{-j\theta}&amp;,\quad 0&lt;\omega&lt;\pi\\
e^{j\theta}&amp;,\quad-\pi&lt;\omega&lt;0\end{cases}\tag{1}$$

Using the sign function $\text{sign}(\omega)$, the frequency response $(1)$ can be rewritten as

$$H(e^{j\omega})=\cos\theta-j\,\text{sign}(\omega)\sin\theta,\quad -\pi&lt;\omega &lt;\pi\tag{2}$$

With the DTFT correspondence

$$-j\,\text{sign}(\omega)\Longleftrightarrow g[n]=\begin{cases}\frac{2}{\pi n},&amp;\quad n\text{ odd}\\
0,&amp;\quad n\text{ even}\end{cases}\tag{3}$$

the impulse response corresponding to $H(e^{j\omega})$ is given by

$$h[n]=\cos\theta\cdot\delta[n]+\sin\theta\cdot g[n]\tag{4}$$

where $g[n]$ is the sequence on the right-hand side of Eq. $(3)$, which is the impulse response of an ideal discrete-time Hilbert transformer. Eq. $(4)$ shows that an ideal phase shifter can be implemented as a weighted parallel connection of a wire ($\cos\theta\cdot\delta[n]$) and a Hilbert transformer.

So your problem can be solved by using any of the many available designs of discrete-time Hilbert transformers. Note that you can get much better performance for a given filter order by taking into account that the approximation needs only be accurate in the given frequency band. For a frequency-domain design method this just means that in the formulation of the desired response given in $(1)$ you replace the positive frequencies by the frequency interval of interest and leave the rest as a "don't care" region. The same is done for the negative frequencies.