The answer is no.
In fact, let $\Gamma \subset \mathbb{P}^3$ be the set of $10$ nodes of a general quartic symmetroid. Then the Gale transform $\Gamma' \subset \mathbb{P}^5$ of $\Gamma$ must be contained in a Veronese surface.
Another characterization is the following: the set $\Gamma$ arises as a linear section of the secant variety $\textrm{Sec}(C_6)$, where $C_6 \subset \mathbb{P}^6$ is a rational normal curve. Note that $\textrm{Sec}(C_6)$ has precisely degree $10$, since $10$ is the number of nodes of a general projection of $C_6$ to $\mathbb{P}^2$.
For (many) more details, see
D. Eisenbud, S. Popescu: The projective geometry of the Gale transform, Journal of Algebra 230 (2000), 127-173,
in particular Example 6.3 p. 153.