Answer
See the detailed answer below.
Work Step by Step
$$\color{blue}{\bf [a]}$$
The diagram below shows all possible orientations of the angular momentum vector $ \vec{L} $ for $ l = 3 $, with each vector labeled by the corresponding $ m $ value. Each vector originates from the same point and is placed on cones with the z-axis as the axis of rotation.
There two copies of the diagram, one is 3D and the other is side view 2D.
$$\color{blue}{\bf [b]}$$
The angle between $ \vec{L} $ and the z-axis is given by:
$$
\theta = \cos^{-1} \left( \frac{L_z}{L} \right)
$$
where $ L_z = m\hbar $ and $ L = \sqrt{l(l+1)}\hbar $.
For the minimum angle, $ m = l $, so:
$$
\theta_{\text{min}} = \cos^{-1} \left( \frac{3\hbar}{\sqrt{3(3+1)}\hbar} \right)
$$
$$
\theta_{\text{min}} = \cos^{-1} \left( \frac{3}{\sqrt{12}} \right) = \cos^{-1} \left( \frac{3}{2\sqrt{3}} \right) \approx \color{red}{\bf 30}^\circ
$$
