Before revision:   We improve this by showing that if r is not a prime power and N:=(d+1)r-r\Big\lceild+2{r+1}\Big\rceil-2, then there is an almost r-embedding \Delta_N\to\mathbb R^d.
After revision:  We improve this further (for d large compared to r): If r is not a prime power and N:=(d+1)r-r\Big\lceild+2{r+1}\Big\rceil-2, then there is an almost r-embedding \Delta_N\to\mathbb R^d.