Let $X$ be a smooth projective variety over $\mathbb C$ of dimension $n+1$. If $Y$ is a smooth very ample hypersurface, we know that except $H^n(X;\mathbb Q)\rightarrow H^n(Y;\mathbb Q)$, the restriction of cohomology (rational coefficients) at other degrees are surjective. My question is the following.

What are the restrictions we can put on $X$, so that there exists a smooth very ample hypersurface $Y$ such that the restriction $H^*(X;\mathbb Q)\rightarrow H^*(Y;\mathbb Q)$ is surjective on all degrees. For example, I would love to know whether it is true if $X$ is a smooth projective toric variety (or with more restrictions). We can replace very ample by ample if it's better. Boundary divisors(torus invariant divisors) satisfies the cohomology requirements, but they might not even be ample. Other possible conditions for $X$ are welcomed as well.

(Should this question be made into community wiki?)

p.s.

My motivation is that in my project, I want to find an ample hypersurface all of whose cohomology classes can be lifted to the ambient space. But uniqueness is not needed. I realized such a hypersurface may not be found for just any variety (e.g., hypersurface in $\mathbb P^n$ if the degree is sufficiently large). So I want to put a reasonable condition on $X$. I have tried to decompose $H^n(Y;\mathbb Q)$ into $H^n(Y;\mathbb Q)=H^n_{fix}(Y;\mathbb Q)\oplus H^n_{var}(Y;\mathbb Q)$ where $H^n_{fix}(Y;\mathbb Q)$ is the image of the restriction from $X$, and $H^n_{var}(Y;\mathbb Q)$ is generated by the vanishing cycles in a Lefschetz pencil. I kind of feel that requiring the monodromy to be trivial does not simplify the problem. But I also realized that if the monodromy $H^n_{var}(Y;\mathbb Q)$ is "very big", I will be equally happy in my project. But what I want for "very big" is quite a messy condition. Any discussion along this line is also welcomed in the comment because there are definitely nice results about monodromy that I'm not aware about.