  It was recently found that excited states of semi-vortex and mixed-mode
solitons are unstable in spin-orbit-coupled Bose-Einstein condensates (BECs)
with contact interactions. We demonstrate a possibility to stabilize such
excited states in a setting based on repulsive dipole-dipole interactions
induced by a polarizing field, oriented perpendicular to the plane in which the
dipolar BEC is trapped. The strength of the field is assumed to grow in the
radial direction $\sim $ $r^{4}$. Excited states of semi-vortex solitons have
vorticities $S$ and $S+1$ in their two components, each being an eigenstate of
the angular momentum. They are fully stable up to $S=5$. Excited state of
mixed-mode solitons feature interweaving necklace structures with opposite
fractional values of the angular momentum in the two components. They are
stable if they are built of dominant angular harmonics $\pm S$, with $S\leq 4$.
Characteristics and stability of these two types of previously unknown
higher-order solitons are systematically analyzed. Their characteristic size is
$\sim 10$ $\mathrm{\mu }$m, with the number of atoms $\lesssim 10^{5}$
