A theorem in Manin's book on cubic forms leads to the conclusion that the least degree of a rational mapping from $\mathbb{P}^2$ to general cubic surfaces(e.g. Picard rank=1) over $\mathbb{Q}$ is 6.

There are two known ways to construct such a mapping: find out two plane rational curves over $\mathbb{Q}$, join each pair of rational points on the two curves by line; or find out two plane rational curves conjugate over some quadratic extension $K$ of $\mathbb{Q}$, join each pair of conjugate points over $K$ on the two curves by line.

Question: Is there any essentially different way to construct degree six mapping over $\mathbb{Q}$?