There are Salem numbers with trace $-3$ and every degree at least $34$

We prove that there exist Salem numbers with trace $-3$ and every even degree $\ge 34$. Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace $-3$ and degree $\le 30$, our result is optimal up to possibly the single value $32$, for which it is expected there are no such numbers.