Your question is related to the problem of the existence of non-trivial extensions of subvarieties $X \subset \mathbb P^N$ to $\mathbb P^{N+1}$. An extension of $X$ is just a subvariety $Y$ of $\mathbb P^{N+1}$ such that $X= Y \cap \mathbb P^{N}$. It is called trivial if $Y$ is the join of $X$ and a point outside of $\mathbb P^N$.
This is a classical question that was studied by the Italian school of algebraic geometry. For instance, Scorza proved that the Veronese surface in $\mathbb P^5$ does not admit non-trivial extensions.
Theorem. Suppose $X$ is not $\mathbb P^n$ P^N$nor a quadric. If$\dim X \ge 2$and$H^1(X,TX\otimes \mathcal O_{\mathbb P^N}(-1))=0$then every extension of$X$is trivial. For a very nice introduction to this circle of ideas see the first chapter of the book Projective geometry and formal geometry by L. Badescu. Unfortunately, the relevant Chapter doesn't seem to be available from Google books. Of course this still does not answer your question as the embedding of$X$into$\mathbb P^N$is fixed. 1 Your question is related to the problem of the existence of non-trivial extensions of subvarieties$X \subset \mathbb P^N$to$\mathbb P^{N+1}$. An extension of$X$is just a subvariety$Y$of$\mathbb P^{N+1}$such that$X= Y \cap \mathbb P^{N}$. It is called trivial if$Y$is the join of$X$and a point outside of$\mathbb P^N$. This is a classical question that was studied by the Italian school of algebraic geometry. For instance, Scorza proved that the Veronese surface in$\mathbb P^5$does not admit non-trivial extensions. More recently, the problem has been studied by Zak, S. L'vovsky, L. Badescu, among many others. In Extensions of projective varieties and deformations by S. L'vovsky you will find the following result: Theorem. Suppose$X$is not$\mathbb P^n$nor a quadric. If$\dim X \ge 2$and$H^1(X,TX\otimes \mathcal O_{\mathbb P^N}(-1))=0$then every extension of$X$is trivial. For a very nice introduction to this circle of ideas see the first chapter of the book Projective geometry and formal geometry by L. Badescu. Unfortunately, the relevant Chapter doesn't seem to be available from Google books. Of course this still does not answer your question as the embedding of$X$into$\mathbb P^N\$ is fixed.