Predicate Logic - Natural Deduction; Assumptions about exists-elimination

Question

I am stuck on how to progress with this proof; i cannot see my next move.

The task is to show $S \to \exists x Q(x) \vdash \exists x (S \to Q(x))$ using natural deduction for predicate logic.

My first attempt was the following;

1: $S \to \exists x Q(x) \space \space \space \space \space \space premise$
2: $x_{0} \space \space S \to Q(x_{0}) \space \space assumption$ (start of scoped box, i do not know how to write them here)
3: $\exists x(S \to Q(x)) \space \space \exists x \space introduction \space 2$ (end of scoped box)
4: $\exists x(S \to Q(x)) \space \space \exists x \space elimination \space 1,2-3$

However, i am not so sure that this is correct. The rule for $\exists$ elimination is
$\frac{\exists x \phi \space \space(And \space we \space manage \space to \space infer \space \gamma \space from \space replacing \space x \space in \phi \space with \space a \space fresh \space variable)}{\gamma}$
I do not know how to render the rule here (bonus question), but here's a link to where it is better described (page 16).

In order to eliminate $\exists x$ i must thus have a formula $\exists x \phi$ as my premise, and the other premise as described in the link above. However, my first premise in this case is $S \to \exists x Q(x)$, which makes me think my second move is illegal.

Can anyone explain if my thinking is correct, and point out my next move, or if i am wrong please explain why i am wrong? (Incidentally the link above describes exactly those rules my course litterature allow me to use, so if i could get help in terms of those rules i would be doubly grateful).